Research Article Formal Pseudodifferential Operators in One and Several Variables ... · 2019. 7....

Research ArticleFormal Pseudodifferential Operators in One and SeveralVariables Central Extensions and Integrable Systems

Jarnishs Beltran1 and Enrique G Reyes2

1Centro de Investigacion en Complejidad Social (CICS) Facultad de Gobierno Universidad del Desarrollo 7610658 Santiago Chile2Departamento de Matematica y Ciencia de la Computacion Universidad de Santiago de Chile Casilla 307 Correo 2 Santiago Chile

Correspondence should be addressed to Enrique G Reyes e g reyesyahooca

Received 22 February 2015 Accepted 19 May 2015

Academic Editor Hagen Neidhardt

Copyright copy 2015 J Beltran and E G Reyes This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We review some aspects of the theory of Lie algebras of (twisted and untwisted) formal pseudodifferential operators in oneand several variables in a general algebraic context We focus mainly on the construction and classification of nontrivial centralextensions As applications we construct hierarchies of centrally extended Lie algebras of formal differential operators in one andseveral variables Manin triples and hierarchies of nonlinear equations in Lax and zero curvature form

1 Introduction

This paper is on some cohomological aspects of Lie algebrasof formal pseudodifferential operators in one and severalindependent variables motivated by previous works onalgebras of importance for integrable systems and symplecticgeometry such as the Lie algebra of vector fields on the circleand their deformations [1 2] or the Lie algebra of differentialoperators see for example [3ndash6]

A very well-known infinite-dimensional Lie algebra ofinterest for physics is theVirasoro algebra [7]This algebra is aone-dimensional central extension of the Lie algebra of vectorfields on the circle (also called centerless Virasoro algebra)which also appears naturally in applications We mention asa recent example that the centerless Virasoro algebra can berealized as an algebra of nonlocal symmetries for theCamassa-Holm and Hunter-Saxton equations [8 9]

Now the Lie algebra of vector fields on the circle isincluded naturally in the Lie algebra of differential operatorson the circle This algebra in turn is included in the Liealgebra of formal pseudodifferential operators on the circlewhich has been studied very carefully for example by Dickey[5] and Adler [10] in connection with the algebraic andgeometric theory of the famous Korteweg-de Vries (KdV)equation and other integrable systems Moreover there exist

nontrivial twisted versions of these works related to ldquotwistedrdquoand ldquoquantumrdquo analogs of classical integrable systems see forinstance [11ndash14] It is certainly reasonable to consider formalpseudodifferential operators as a general arena for integrablesystems

Our aim in this work is to study central extensions of Liealgebras of formal pseudodifferential operators in a generalalgebraic setting and to apply this study to the construction ofhierarchies of centrally extended Lie algebras Manin triples[15] and nonlinear integrable equations in one and severalindependent variables

Wemention four examples of relevant central extensionsCenterless Virasoro has a unique central extension see [3]Also a 2-cocycle of the algebra of differential operators on thecircle was constructed in [6] and a 2-cocycle of the algebra ofpseudodifferential operators on the circle was constructed byKravchenko andKhesin using logarithms see [16 17] Finallya 2-cocycle for a quantum analog of the algebra of pseu-dodifferential operators was considered in [11]

In this paper we consider algebraic versions of theseresults and we present classifications of central extensions Inparticular we show thatmany of the constructions of cocyclesappearing in the literature (see for instance [17] or [4]) arevalid well beyond their original framework

We have divided our work in three main sections

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 210346 16 pageshttpdxdoiorg1011552015210346

2 Advances in Mathematical Physics

In Section 2 we introduce the main objects considered inthis work We define formal pseudodifferential operators (orldquopseudodifferential symbolsrdquo) in one variable on an arbitraryassociative and commutative algebra and we construct theKravchenko-Khesin logarithmic 2-cocycle in this generalcontext We also construct a hierarchy of centrally extendedLie algebras of differential operators via the logarithmic 2-cocycle This construction generalizes a theorem by Khesin[18] on hierarchies of Lie algebras of differential operators onthe circle Finally motivated by [11 19 20] and the later paper[12] we consider twisted pseudodifferential symbols on arbi-trary associative and commutative algebras and in analogywith the untwisted case we construct central extensions andhierarchies of twisted centrally extended algebras

In Section 3 we consider pseudodifferential symbols inseveral independent variables on an arbitrary associativeand commutative algebra Our main motivation for con-sidering the several variables case in detail is the relativeabsence of examples of integrable equations in this contextIndeed besides the equations of the standard KP hierarchy[5] and their cousins there are not very many generalconstructions of integrable equations in several independentvariables Important exceptions are the equations introducedby Tenenblat and her coworkers (see [21] and referencestherein) and the hierarchies considered by Parshin [22]We construct central extensions in the several variables caseusing logarithmic cocycles and we also exhibit hierarchies ofLie algebras of differential operators in several independentvariables admitting central extensions We then considerthe work [23] by Dzhumadilrsquodaev in which he presents aclassification of central extensions The paper [23] is quitetechnical and it requires a very careful and critical readingand so we have decided to explain how to prove the mainresult of [23] using an inductive argument We present afull inductive proof of Dzhumadilrsquodaevrsquos theorem using sometechnical homological tools elsewhere see [24]

Finally in Section 4we introduceManin triples we definedouble extensions for the algebras of (twisted) pseudodiffer-ential symbols in one and several independent variables andusing a general algebraic theorem [25] we construct Manintriples for these algebras thereby putting [17 26ndash29] in avery general framework We also apply our algebraic resultsto the construction of integrable systems in one and severalindependent variables roughly following the techniques of[4 11 22]

2 The Algebra of Formal PseudodifferentialSymbols in One Variable

21 Basic Definitions and Preliminary Results Let A be anassociative and commutative algebra and let 120575 A rarr A

be a derivation on A that is 120575 is a linear map such that120575(119886119887) = 119886120575(119887) + 120575(119886)119887 for all 119886 119887 isin A The algebra of formaldifferential symbolsDO is generated byA and a symbol 120585withthe relation

120585 ∘ 119886 = 119886120585 + 120575 (119886) (1)

for all 119886 isin AThe algebraA is a subalgebra of DO and we canprove inductively that

120585119899

∘ 119886 =

119899

sum

119895=0(119899

119895) 120575

119895

(119886) 120585119899minus119895 (2)

for all 119886 isin A and 119899 ge 0We extend the algebra DO to obtain the algebra of pseu-

dodifferential symbols ΨDO by introducing differentiationswith negative exponents A general element of ΨDO is aformal series 119875 of the form

119875 =

119899

sum

minusinfin

119886119894120585119894 where 119886

119894isin A (3)

We set

120585minus1∘ 119886 =

infin

sum

119894=0(minus1)119894 120575119894 (119886) 120585minus1minus119894 (4)

and so (2) generalizes to

120585119899

∘ 119886 =

infin

sum

119895=0(119899

119895) 120575

119895

(119886) 120585119899minus119895 (5)

for all 119899 isin Z in which the binomial coefficient is defined by

(119899

119895) =

119899 (119899 minus 1) sdot sdot sdot (119899 minus 119895 + 1)119895

(6)

for 119899 119895 isin Z If 1198751 = sum119872

119894=minusinfin119886119894120585119894 and 1198752 = sum

119873

119895=minusinfin119887119895120585119895 then

1198751 ∘ 1198752 =infin

sum

119896=0

1119896

120597119896

120597120585119896(1198751) 120575

119896

(1198752) (7)

The Lie algebra structure on ΨDO is given by the usualcommutator [119860 119861] = 119860 ∘ 119861 minus 119861 ∘ 119860 so that for instance

[120585120572

119886120585119899

] =

infin

sum

119895=1(119899

119895) 120575

119895

(119886) 120585120572+119899minus119895

(8)

Lemma 1 For any nonnegative integer 119898 and 119886 119887 isin A wehave

120575119898

(119886119887) =

119898

sum

119895=0(119898

119895)120575

119898minus119895

(119886) 120575119895

(119887) (9)

Proof Proof is by induction

Let 120591 A rarr C be a 120575-invariant trace on A that is 120591 isa linear map satisfying 120591(119886119887) = 120591(119887119886) and 120591(120575(119886)) = 0 for all119886 119887 isin A For example if A = Diff(1198781) 120575 = 119889119889119909 the linearfunctional 120591 Diff(1198781) rarr C given by 120591(119891) = int

1198781 119891(119909)119889119909 is a

120575-invariant trace

Lemma 2 Let A be an algebra 120575 a derivation on A 120591 a 120575-invariant trace on A and 119898 a positive integer Then we canldquointegrate by partsrdquo that is for all 119886 119887 isin A we have

120591 (119887120575119898

(119886)) = (minus1)119898 120591 (119886120575119898 (119887)) (10)

Advances in Mathematical Physics 3

Proof We have that 120575(119886119887) = 119886120575(119887) + 120575(119886)119887 and so 0 =

120591(120575(119886119887)) = 120591(119886120575(119887)) + 120591(120575(119886)119887) then minus120591(119886120575(119887)) = 120591(120575(119886)119887)We now proceed by induction on119898

Proposition 3 LetA be an algebra 120575 a derivation onA and120591 a 120575-invariant trace Then the linear map res Ψ119863119874 rarr C

defined by

res(119872

sum

119894=minusinfin

119886119894120585119894

) = 120591 (119886minus1) (11)

is a trace onΨ119863119874 that is res is linear and it satisfies res(119860119861) =res(119861119860) for all 119860 119861 isin Ψ119863119874 This is the Adler-Manin non-commutative residue introduced in [10 30]

Proof We use the elementary identity

(119886

119887) = (minus1)119887 (

119887 minus 119886 minus 1119887

) (12)

for 119886 lt 0 and 119887 ge 0 Since res is linear it is sufficient to showthat for any 119886 119887 isin A and119898 119899 isin Z

res (119886120585119899 ∘ 119887120585119898) = res (119887120585119898 ∘ 119886120585119899) (13)

We consider several casesLet119898 sdot 119899 ⩾ 0

res (119886120585119899 ∘ 119887120585119898) =infin

sum

119895=0res((

119899

119895) 119886120575

119895

(119887) 120585119898+119899minus119895

)

= 120591((119899

119898 + 119899 + 1)119886120575

119898+119899+1(119887))

(14)

and this is identically zero since we obtain the coefficient of120585minus1 when 119895 = 119898 + 119899 + 1 and ( 119899

119898+119899+1 ) = 0 Similarly

res (119887120585119898 ∘ 119886120585119899) = 120591((119898

119898 + 119899 + 1)119886120575

119898+119899+1(119887)) = 0 (15)

Now we let119898 119899 lt 0 then119898+119899minus119895 le 119899+119898 lt minus1 (119895 a positiveinteger) and

res (119886120585119899 ∘ 119887120585119898) = res(infin

sum

119895=0(119899

119895) 119886120575

119895

(119887) 120585119898+119899minus119895

) (16)

But as 119898 + 119899 minus 119895 lt minus1 the coefficient of 120585minus1 is 0 and thenres(119886120585119899119887120585119898) = 0 Analogously we find res(119887120585119898 ∘ 119886120585119899) = 0

Now assume that 119899 ⩾ 0 119898 lt 0 If 119899 + 119898 lt minus1 then

res (119886120585119899 ∘ 119887120585119898) = res(infin

sum

119895=0(119899

119895) 119886120575

119895

(119887) 120585119898+119899minus119895

) = 0 (17)

because 119899 + 119898 minus 119895 lt minus1 minus 119895 le minus1 and then 119899 + 119898 minus 119895 lt minus1Analogously res(119887120585119898 ∘ 119886120585119899) = 0

Finally we let 119899 + 119898 ⩾ minus1 set 119896 = 119898 + 119899 Then

res (119886120585119899 ∘ 119887120585119898) = 120591((119899

119896 + 1)119886120575

119896+1(119887))

= (119899

119896 + 1)120591 (119886120575

119896+1(119887))

(18)

and on the other hand (10) and (12) imply

res (119887120585119898 ∘ 119886120585119899) = 120591((119898

119896 + 1)119887120575

119896+1(119886))

= (119896 minus 119899

119896 + 1)120591 (119887120575

119896+1(119886))

= (minus1)119896+1 (119899

119896 + 1)120591 (119887120575

119896+1(119886))

= (minus1)119896+1 (minus1)119896+1 (119899

119896 + 1)120591 (119886120575

119896+1(119887))

= (119899

119896 + 1)120591 (119886120575

119896+1(119887))

(19)

Remark 4 Our definition of residue follows the conventionsof [23] For example if A = Diff(1198781) 120575 = 119889119889119909 and 120591

Diff(1198781) rarr C is given by 120591(119891) = int1198781 119891(119909)119889119909 then with our

convention

res (119860) = res(119899

sum

119894=minusinfin

119886119894(119909) 120597

119894

) = 120591 (119886minus1 (119909)) (20)

which is slightly different from the notation used in [4 16ndash18]

Corollary 5 The bilinear form ⟨119860 119861⟩ = res(119860 ∘ 119861) is 119886119889-invariant that is it satisfies ⟨[119860 119861] 119862⟩ = ⟨119860 [119861 119862]⟩

Now if 119875 = 119886119899120585119899

+ 119886119899minus1120585

119899minus1+ sdot sdot sdot is a pseudodifferential

symbol such that 119886119899= 0 and 119886

119898= 0 for all119898 gt 119899 we say that

119899 is the order or degree of 119875 The following observation hasresulted to be fundamental for the theory see for instance [510 17 31] The algebra ΨDO can be decomposed as a (vectorspace) direct sum ΨDO = DO oplus INT where

DO =

119899

sum

119894=0119886119894120585119894

| 119886119894isinA

INT = minus1sum

119894=minusinfin

119886119894120585119894

| 119886119894isinA

(21)

Proposition 6 The subalgebras119863119874 (of differential operators)and 119868119873119879 (of pseudodifferential symbols of order le minus1) areisotropic subspaces of Ψ119863119874 with respect to the bilinear formdefined in Corollary 5 that is the restrictions of this form toboth119863119874 and 119868119873119879 vanish

22 On Cohomology of Lie Algebras Having reviewed theelementary properties of ΨDO we now summarize somebasic facts on the cohomology of Lie algebras in order tofix our notation We will study the cohomology of ΨDO inSection 23


221 Basic Definitions Suppose that g is a Lie algebra andthat 119860 is a module over g A 119902-dimensional cochain of thealgebra g with coefficients in 119860 is a skew-symmetric 119902-linearfunctional on gwith values in119860 the space of all such cochainsis denoted by 119862119902(g 119860) and we also set 1198620

(g 119860) = 119860 Thedifferential 119889 = 119889

119902 119862

119902

(g 119860) rarr 119862119902+1(g 119860) is defined by

the formula

119889119888 (1198921 119892119902+1) = sum

1le119904lt119905le119902+1(minus1)119904+119905minus1

sdot 119888 ([119892119904 119892

119905] 1198921 119892119904 119892119905 119892119902+1)

+ sum

1le119904le119902+1(minus1)119904 119892

119904119888 (1198921 119892119904 119892119902+1)

(22)

where 119888 isin 119862(g 119860) and1198921 119892119902+1 isin gWe also set119862119902(g 119860) =0 for 119902 lt 0 and 119889

119902= 0 for 119902 lt 0 We can check that

119889119902+1 ∘ 119889119902 = 0 for all 119902 and therefore 119862119902(g 119860) 119889

119902isinZ is analgebraic complexThis complex is denoted by119862∙(g 119860) while119867119902

(g 119860) denotes the 119902-cohomology space of the algebra gwith coefficients in 119860 If 119860 is a trivial g-module then thesecond sum of in the right-hand side of formula (22) vanishesand it may be ignored If 119860 is a field the notations 119862119902(g 119860)119867119902

(g 119860) are abbreviated to 119862119902(g)119867119902

(g)

222 Algebraic Interpretations of Cohomology A derivation120575 of the Lie algebra g is a linear map 120575 g rarr g such that120575([119909 119910]) = [120575(119909) 119910] + [119909 120575(119910)] A derivation is inner if 120575 =120575119909(sdot) = [119909 sdot] where 119909 isin g is a fixed elementOuter derivations

are by definition elements of the quotient space of all deriva-tions module the subspace of inner derivations The proof ofthe following proposition is in [32] Chapter 1 Section 4

Proposition 7 1198671(g g) can be interpreted as the space of

outer derivations of the algebra g

Definition 8 A central extension of a Lie algebra g by a vectorspace n is a Lie algebra g whose underlying vector spaceg = g oplus n is equipped with the following Lie bracket

[(119883 119906) (119884 V)] = ([119883 119884] 119888 (119883 119884)) (23)

for some bilinear map 119888 g times g rarr n

Note that 119888 depends only on119883 and 119884 but not on 119906 and VThis implies that n is the center of the Lie algebra g

The skew-symmetry bilinearity and the Jacoby identityon the Lie algebra g are equivalent to the antisymmetrybilinearity and the following 2-cocycle identity for the map119888

119888 ([119883 119884] 119885) + 119888 ([119885119883] 119884) + 119888 ([119884 119885] 119883) = 0 (24)

for any119883119884 119885 isin gTwo 2-cocycles 119888 on g with values in n differing by a 2-

coboundary give rise to isomorphic central extensions of gWe have the following result see [32 page 33]

Proposition 9 There is a one-to-one correspondence betweenequivalence classes of central extensions of g by n and elementsof1198672

(gn)

The referee has pointed out that Proposition 9 combinedwith Proposition 10 below allows us to effectively calculatecentral extensions

Proposition 10 Let g be a Lie algebra The space of one-dimensional central extensions 1198672

(gC) is isomorphic to asubspace of the first cohomology space 1198671

(g glowast) Specificallyif we denote by 1198671

(g) the subspace of 1198671(g glowast) generated by

cohomology classes of cocycles 120595 g rarr glowast such that

120595 (119909) sdot 119909 = 0 (25)

for 119909 isin g then1198672(gC) cong 119867

1(g)

This general proposition is due to Dzhumadilrsquodaev whoused it in [33] (in the case of one-dimensional central exten-sions induced by fields of characteristic 119901 gt 0) and in [34] (inthe zero characteristic case) for the study of central extensionsof Lie algebras of Cartan type see for example [33ndash35]Proposition 10 has the following corollary also pointed outby the referee (see also [23 33])

Corollary 11 If the Lie algebra g admits an invariant symmet-ric and nondegenerate bilinear form ⟨ ⟩ then the space of one-dimensional central extensions 1198672

(gC) is isomorphic to thespace of outer derivations 120575 g rarr g such that ⟨120575(119909) 119909⟩ = 0for all 119909 isin g

Proof Theexistence of ⟨ ⟩ allows us to identify1198671(g g)with

1198671(g glowast) Now if we take120595 so that the cohomology class of120595

is in1198671(g) we obtain a set of derivations of the form 120575

120595+[119886 sdot]

for some 119886 isin g Then invariance of ⟨ ⟩ implies that 120595 isin

1198671(g) if and only if ⟨120575

120595(119909) 119909⟩ = 0 for all 119909 isin g

23 Outer Derivations and Central Extensions of Ψ119863119874 Wegoback to the algebraΨDOconsidered in Section 21 Follow-ing [16] we write formally the identity 120585119905 = 119890

119905 log 120585 Thisimplies that

119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120585119905

= log 120585 (26)

Hence setting 120572 = 119905 in (8) and differentiating at 119905 = 0 using(119889119889119905)|

119905=0 (119905

119895) = (minus1)119895+1119895 we obtain

[log 120585 119886120585119899] =infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 120585119899minus119895

(27)

It follows that if 119875 isin ΨDO then [log 120585 119875] is also an elementof ΨDO even though log 120585 itself is not

In the next proposition we will make use of the followingcombinatorial identity (see for instance [12])

Lemma 12 Let 119904 ge 1 and 119896 ge 0 be integers and 120573 isin Z Then

(minus1)119904minus1

119904(120573 minus 119904

119896) =

119904+119896

sum

119895=119904


119895(

120573

119904 + 119896 minus 119895)(

119895

119895 minus 119904) (28)


Proposition 13 and Theorem 14 below were proved byKravchenko and Khesin [16] in the caseA = Diff(1198781)

Proposition 13 [log 120585 sdot] defines a (resp an outer) derivationof the associative (resp Lie) algebra Ψ119863119874

Proof We note that the proposition does not follow fromthe fact that for any associative algebra the map 119886 997891rarr [119886 sdot]

determines a derivation since in our case log 120585 is not anelement of ΨDO

First of all it is not difficult to see using (27) that [log 120585 119876]belongs to ΨDO for any 119876 isin ΨDO Now assuming that[log 120585 sdot] is a derivation it is trivial to prove that it is outerderivation of the Lie algebra ΨDO if [log 120585 sdot] = [119860 sdot] forsome119860 isin ΨDO then log 120585minus119860 belongs to the center ofΨDOand so log 120585 isin ΨDO a contradiction

We show that [log 120585 sdot] is a derivation It is sufficient toprove that for any 119886 119887 isin 119860 119898 119899 isin Z

[log 120585 119886120585119899 ∘ 119887120585119898] = [log 120585 119886120585119899] ∘ 119887120585119898 + 119886120585119899

∘ [log 120585 119887120585119898] (29)

Indeed for the left side of (29) we have

[log 120585 119886120585119899 ∘ 119887120585119898] = [log 120585infin

sum

119896=119900

(119899

119896)119886120575

119895

(119887) 120585119899+119898minus119896

]

=

infin

sum

119896=119900

infin

sum

119895=1(119899

119896)(minus1)119895+1

119895120575119895

(119886120575119896

(119887)) 120585119899+119898minus119896minus119895

(30)

On the other hand for the right side of (29) we have

[log 120585 119886120585119899] ∘ 119887120585119898 + 119886120585119899 ∘ [log 120585 119887120585119898]

= (

infin

sum

119895=119900

(minus1)119895+1

119895120575119895

(119886) 120585119899minus119895

)119887120585119898

+ 119886120585119899

(

infin

sum

119895=119900

(minus1)119895+1

119895120575119895

(119887) 120585119898minus119895

)

=

infin

sum

119895=1

infin

sum

119896=0(119899 minus 119895

119896)(minus1)119895+1

119895120575119895

(119886) 120575119896

(119887) 120585119899+119898minus119896minus119895

+

infin

sum

119895=1

infin

sum

119896=0(119899

119896)(minus1)119895+1

119895119886120575

119896

(120575119895

(119887)) 120585119899+119898minus119896minus119895

=

infin

sum

119895=1

infin

sum

119896=0(119899 minus 119895

119896)(minus1)119895+1

119895120575119895

(119886) 120575119896

(119887) 120585119899+119898minus119896minus119895

+

infin

sum

119895=1

infin

sum

119896=0(119899

119896)(minus1)119895+1

119895119886120575

119896+119895

(119887) 120585119899+119898minus119896minus119895

(31)

Now for any integer 119903 ⩾ 1 the coefficient of 120585119899+119896minus119903 in (30) is

sum(119899

119896)(minus1)119895+1

119895120575119895

(119886120575119896

(119887)) (32)

where the summation is over all integers 119895 ⩾ 1 119896 ⩾ 0 suchthat 119895 + 119896 = 119903 Using (9) we have

sum

119895

sum

119894=119900

(119899

119896)(

119894

119895)(minus1)119895+1

119895120575119895minus119894

(119886) 120575119894+119896

(119887)

= sum

119895minus1

sum

119894=119900

(119899

119896)(

119894

119895)(minus1)119895+1

119895120575119895minus119894

(119886) 120575119894+119896

(119887)

+sum(119899

119896)(minus1)119895+1

119895119886120575

119903

(119887)

(33)

where both summations are over all integers 119895 ⩾ 1 119896 ⩾ 0 suchthat 119895 + 119896 = 119903 On the other hand for 119903 ⩾ 1 the coefficient120585119899+119896minus119903 of (31) is

sum(119899 minus 119904

119897)(minus1)119904+1

119904120575119904

(119886) 120575119897

(119887)

+sum(119899

119897)(minus1)119904+1

119904119886120575

119903

(119887)

(34)

where the sum is over all integers 119904 ⩾ 1 119897 ⩾ 0 such that 119904+ 119897 =119903 Therefore (33) and (34) are equal if for fixed integers 119904 119897 asabove we have

sum(119899

119896)(

119895

119894)(minus1)119895+1

119895= (

119899 minus 119904

119897)(minus1)119904+1

119904 (35)

where the sum is over all integers 119895 ⩾ 1 119896 ⩾ 0 and 119894 ⩾ 0 suchthat 119894 = 119895 minus 119904 119894 + 119896 = 119897 This amounts to showing that

119897+119904

sum

119895=119904

(119899

119897 + 119904 minus 119895)(

119895

119895 minus 119904)(minus1)119895+1

119895= (

119899 minus 119904

119897)(minus1)119904+1

119904 (36)

and this is consequence of (28)

Theorem 14 The map 119888 Ψ119863119874 times Ψ119863119874 rarr C given by

119888 (119860 119861) = res ([log 120585 119860] ∘ 119861) (37)

defines a Lie algebra 2-cocycle on Ψ119863119874

Proof It is easy to see that res([log 120585 119886120585119899]) = 0 for 119899 le minus1while for 119899 ge 0 we have

res ([log 120585 119886120585119899]) = 120591 ((minus1)119899

119899 + 1120575119899+1

(119886)) = 0 (38)

and so res([log 120585 119875]) = 0 for all 119875 isin ΨDO It follows that

119888 ([119861 119860]) = res ([log 120585 119861]119860)

= res ([log 120585 119861119860] minus119861 [log 120585 119860])

= minus 119888 (119860 119861)

(39)

and so 119888 is skew-symmetric It remains to prove the cocycleidentity (24) This a direct calculation using Corollary 5 andthe fact that [log 120585 sdot] is a Lie algebra derivation


Remark 15 In the case of the Lie algebra of pseudodifferentialsymbols on 1198781 the restriction of the cocycle (37) to the sub-algebra of vector fields on 1198781 is the Gelfand-Fuchs cocycle

119888 (119891 (119909) 120597 119892 (119909) 120597) =16int119891

1015840

(119909) 11989210158401015840

(119909) 119889119909 (40)

see for instance [17 Prop 412] This cocycle is nontrivialsee [3] and therefore cocycle (37) is nontrivial [16] It is alsoknown (see for instance [17]) that the restriction of 2-cocycle(37) to the subalgebra of differential operator DO(1198781) sub

ΨDO(1198781) is a multiple of the Kac-Peterson cocycle [6]

119888 (119891120597119899

119892120597119898

) =119899119898

(119898 + 119899 + 1)int1198781119891(119898+1)

119892(119899)

119889120579 (41)

Interestingly the Lie algebra DO(1198781) has exactly one centralextension [36] but ΨDO(1198781) has two independent centralextensions [17 37] in addition to (37) the following expres-sion defines a nontrivial cocycle

119888 (119860 119861) = int1198781([119909 119860] ∘ 119861) (42)

We will reprove this result as a corollary of our study ofcentral extensions of Lie algebras of formal pseudodifferentialoperators in several variables see Lemma 33 in Section 34

Remark 16 LetΣ be a compact Riemann surface and letM bethe space ofmeromorphic functions onΣ Fix ameromorphicvector field V on Σ and denote by 119863V the operator of Liederivative 119871V along the field V locally if V(119909) = 119891(119909)120597120597119909 and119892 isin M then 119863V(119892) = 119871V(119892(119909)) = 119891(119909)119892

1015840

(119909) The associativealgebra of meromorphic pseudodifferential symbols is (see[4])

119872ΨDO =

119899

sum

119894=minusinfin

119886119894119863V

119894

| 119886119894isinM (43)

with multiplication defined as in (7) We consider theLie algebra structure of 119872ΨDO and the residue mapres

119863(sum

119899

119894=minusinfin119886119894119863119894

V) = 119886minus1119863

minus1V where 119863minus1

V is understood as ameromorphic differential on Σ We further define the traceassociated to the point119875 isin Σ byTr119860 = res

119875(res

119863(119860))Then as

a consequence of Theorem 14 we have a nontrivial 2-cocycleon119872ΨDO given by 119888V(119860 119861) = Tr([log119863V 119860]119861)This cocyclefirst appeared in [4]

24 A Hierarchy of Centrally Extended Lie Algebras ΨDOis not unique in admitting nontrivial central extensions Infact a whole hierarchy of Lie algebras doesThis fact was firstobserved by Khesin [18] in the caseA = Diff(1198781)

For any positive integer 119898 we let ΨDO119898be the subalge-

bra of ΨDO consisting of differential operators of the formsum119872

119894=119898119886119894120585119894 for some nonnegative integer119872

Theorem 17 Suppose that the bilinear form defined inCorollary 5 is nondegenerate on Ψ119863119874 The restriction of the2-cocycle 119888(119860 119861) = res([log 120585 119860]119861) to Ψ119863119874

119898defines a non-

trivial central extension of this subalgebra

Proof Using the bilinear form ⟨ ⟩ defined in Corollary 5we identify ΨDO with the dual space ΨDOlowast and we haveΨDOlowast

119898≃ sum

minus119898minus1119894=minusinfin

119886119894120585119894

119886119894isin A If we assume that 119888(119860 119861)

is a coboundary then (22) implies that for119860 = sum119872

119894=119898119886119894120585119894 119861 =

sum119872

119894=119898119887119894120585119894

isin ΨDO119898 and a fixed 119871 = sum


119897119894120585119894

isin ΨDOlowast

119898we

have

119888 (119860 119861) = res ([119860 119861] 119871) = res([

[

119872

sum

119894=119898

119886119894120585119894

119872

sum

119895=119898

119887119895120585119895]

]

sdot

minus119898minus1sum

119896=minusinfin

119897119896120585119896

)

= res( sum

119894119895119905119896119904119906=1198981198980minusinfin00((coefficient) 120575(119904) (119886

119894)

sdot 120575(119905)

(119887119895) 120575

(119906)

(119897119896) 120585

119894+119895+119896minus(119905+119906+119904)

))

= 120591 (coefficientminus1)

(44)

Then the coefficient of 120585minus1 satisfies

119894 + 119895 + 119896 minus (119905 + 119906 + 119904) = minus 1 (45)

for 119904 119905 119906 ge 0 119896 le minus119898 minus 1 On the other hand we have that

119888 (119860 119861) = res ([log 120585 119860] 119861) = res([log 120585119872

sum

119894=119898

119886119894120585119894

]

sdot

119872

sum

119895=119898

119887119895120585119895

) = res( sum

119894119895119905119904=11989811989800(coefficient) 120575(119904) (119886

119894)

sdot 120575(119905)

(119887119895) 120585

119894+119895minus(119904+119905)

) = 120591 (coefficientminus1)

(46)


119894 + 119895 minus (119904 + 119905) = minus 1 (47)

comparing (45) and (47) for any 119860 119861 and we find 119896 = 119906and this contradicts the condition for 119896 and 119906 appearing after(45)

We note that in this case we cannot use Corollary 11 toprove that the 2-cocycle 119888 determines a nontrivial centralextension because [log 120585 sdot] is not a derivation on ΨDO

119898

25 The Algebra of Twisted Pseudodifferential Symbols in OneVariable We consider the algebra of twisted pseudodifferen-tial symbols and its corresponding logarithmic cocycle fol-lowing [12] Particular examples have appeared much earliersee for instance [11 19 20]


Definition 18 Let 120590 A rarr A be an automorphism of fixedalgebraA and let 119886 119887 isin A

(1) A 120590-derivation on A is a linear map 120575 such that120575(119886119887) = 120575(119886)119887 + 120590(119886)120575(119887)

(2) A 120590-trace onA is a linear map 120591 A rarr C such that120591(119886119887) = 120591(120590(119887)119886)

Given a triplet (A 120575 120590) as above the algebra of twistedformal pseudodifferential symbols ΨDO

120590is the set of all

formal Laurent series in 120585 with coefficients inA

ΨDO120590=

119873

sum

119894=minusinfin

119886119894120585119894

119873 isinZ 119886119899isinA (48)

equipped with a multiplication determined by the rules

120585119886 = 120590 (119886) 120585 + 120575 (119886)

120585120585minus1= 120585

minus1120585 = 1

(49)

For example for each 119899 ge 0 we have

120585119899

119886 =

119899

sum

119894=0119875119894119899(120590 120575) (119886) 120585

119894

(50)

where 119875119894119899(120590 120575) is a noncommutative polynomial in 120590 and 120575

with ( 119899119894) terms of total degree 119899 such that the degree of 120590 is 119894

If 119899 = 2 for instance we get 1205852119886 = 1205752(119886) + (120575120590(119886) + 120590120575(119886))120585 +1205902(119886)120585

2 We extend (50) for 119899 lt 0 We obtain

120585119899

119886 =

infin

sum

1198941=0sdot sdot sdot

infin

sum

119894119899=0(minus1)1198941+sdotsdotsdot+119894119899

sdot 120590minus1(120575120590

minus1)119894119899

sdot sdot sdot 120590minus1(120575120590

minus1)1198941(119886) 120585

119899minus1198941minussdotsdotsdotminus1119899

(51)

and it follows that if 119860 = sum119873

119899=minusinfin119886119899120585119899 and 119861 = sum119872

119898=minusinfin119887119898120585119898

then

119860119861 =

119872

sum

119898=minusinfin

sum

119899lt0sum

119894ge0(minus1)|119894| 119886

119899120590minus1(120575120590

minus1)119894minus119899


minus1)1198941(119887119898) 120585

119898+119899+|119894|

+

119872

sum

119898=minusinfin

119873

sum

119899=0

119899

sum

119895=0119886119899119875119895119899(120590 120575) (119887

119898) 120585

119898+119895

(52)

where 119894 = (1198941 119894119899) is an 119899-tuple of integers and |119894| = 1198941 +sdot sdot sdot + 119894

119899 The next proposition and theorem are proved in [12]

Proposition 19 LetA be algebra 120590 an automorphism ofA 120591a 120590-trace onA and 120575 a 120590-derivation onA If 120591∘120575 = 0 then forany 119886 119887 isin A and any 119898-tuple 119894 = (1198941 119894119898) of nonnegativeintegers we have

120591 (119887120590minus1(120575120590

minus1)1198941sdot sdot sdot 120590

minus1(120575120590

minus1)119894119898

(119886))

= (minus1)1198941+sdotsdotsdot+119894119898 120591 (119886120575119894119898120590120575119894119898minus1 sdot sdot sdot 1205901205751198941 (119887)) (53)

Theorem 20 LetA be algebra 120590 an automorphism ofA 120591 a120590-trace onA and 120575 a 120590-derivation onA If 120591 ∘ 120575 = 0 then thelinear functional res Ψ119863119874

120590rarr C defined by

res(119899

sum

119894=minusinfin

119886119894120585119894

) = 120591 (119886minus1) (54)

is a trace on Ψ119863119874120590

As pointed out in [12] if 120575 ∘ 120590 = 120590 ∘ 120575 formulae (50) and(51) simplify to

120585119899

119886 =

infin

sum

119895=0(119899

119895) 120575

119895

(120590119899minus119895

(119886)) 120585119899minus119895

(55)

For example the twisted pseudodifferential operators consid-ered in [11] satisfy (55) We introduce a twisted logarithmiccocycle assuming that 120575 and 120590 commute Let 120590

119905be a 1-

parameter group of automorphisms of A with 1205901 = 120590 Weformally replace the integer 119899 by 119905 isin R in (55) and obtain

120585119905

119886 minus 120590119905(119886) 120585

119905

=

infin

sum

119895=1(119905

119895) 120575

119895

(120590119905minus119895) 120585

119905minus119895

(56)

Taking derivatives with respect to 119905 at 119905 = 0 as in Section 23we obtain the commutation relation

[log 120585 119886] = 119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120590119905(119886) +

infin

sum

119895=1


119895120590minus119895

120575119895

(119886) 120585minus119895

(57)

We note that [log 120585 120585] = 0 The following two results are alsoproven in [12]

Proposition 21 The map [log 120585 sdot] Ψ119863119874120590

rarr Ψ119863119874120590

defined by

[log 120585 119886120585119899] = 119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120590119905(119886) 120585

119899

+

infin

sum

119895=1


119895120590minus119895

120575119895

(119886) 120585minus119895

(58)

is a derivation

Theorem 22 The 2-cochain 119888(119860 119861) = res([log 120585 119860]119861) is a Liealgebra 2-cocycle

Now we go beyond [12] The algebra Ψ120590has a direct sum

decomposition as a vector space

ΨDO120590= DO

120590oplus INT

120590 (59)

where

DO120590=

119899

sum

119894=0119886119894120585119894

| 119886119894isinA

INT120590=

minus1sum

119894=minusinfin

119886119894120585119894

| 119886119894isinA

(60)


Moreover we can prove a result analogous to Proposition 6and we can also produce hierarchies of centrally extendedalgebras of twisted pseudodifferential symbols as inTheorem 17

Proposition 23 res([119860 119861]) = 0 for all 119860 119861 isin Ψ119863119874120590 This

implies that the bilinear form ⟨119860 119861⟩120590= res(119860119861) is invariant

that is it satisfies ⟨[119860 119861] 119862⟩120590= ⟨119860 [119861 119862]⟩

120590 Also the sub-

algebras 119863119874120590and 119868119873119879

120590are isotropic subspaces of Ψ119863119874 that

is the restrictions of the form ⟨ ⟩120590to both 119863119874

120590and 119868119873119879

120590

vanish

Theorem 24 Suppose that the bilinear form defined inProposition 23 is nondegenerate on Ψ119863119874

120590 The restriction of

the 2-cocycle 119888(119860 119861) = res([log 120585 119860]119861) to Ψ119863119874120590119898

defines anontrivial central extension of this subalgebra

Proof In addition to DO120590and INT

120590 we note that it is also

possible to define subalgebras ΨDO120590119898

of ΨDO120590by

ΨDO120590119898

=

119872

sum

119894=119898

119886119894120585119894

| 119886119894isinA (61)

The proof now follows along the lines of the demonstrationof Theorem 17

3 The Algebra of Formal PseudodifferentialSymbols in Several Variables

31 Preliminaries Our notation mainly follows [23] We fix119899 isin Z+ and we set

Γ119899= 120572 = (1205721 120572119894 120572119899) 120572119894 isinZ

Γ+

119899= 120572 isin Γ

119899 120572

119894isinZ

+

1le 119894 le 119899

120576119894= (0 0 1 0 0)

120599 =

119899

sum

119894=1120576119894isin Γ

119899

120572 =

119899

prod

119894=1120572119894 120572

119894isin Γ

+

119899

(62)

Also 120585120572 = 12058512057211 sdot sdot sdot 120585120572119899

119899 120575120574(119887) = 12057512057411 ∘ sdot sdot sdot ∘ 120575

120574119899

119899(119887) for linear maps

1205751 120575119899 and

(120572

120574) = prod

119894

(120572119894

120574119894

) (63)

in which the binomial coefficient is defined as in Section 2Let A be an algebra on C and let 120575

119894 with 119894 = 1 119899

be (commuting) derivations on A The algebra of formaldifferential symbols in several variables DO

119899is by definition

the algebra generated byA and symbols 120585119894with the relations

120585119894119886 = 119886120585

119894+ 120575

119894(119886) (64)

for all 119886 isin A and 119894 = 1 119899 Elements of DO119899are of the

form 119860 = sum120572isinΓ+

119899

119886120572120585120572 Using (64) we can prove that

119886120585120572

sdot 119887120585120573

= sum

120574isinΓ+

119899

(120572

120574)119886120575

120574

(119887) 120585120572+120573minus120574

(65)

We extend the algebraDO119899to the algebraΨ

119899DOof formal

pseudodifferential operators by introducing differentiationswith negative exponents via

120585minus1119894119886 = sum

119895ge0(minus1)119895 120575119895

119894(119886) 120585

minus1minus119895119894

(66)

andwe define a structure of Lie algebra onΨ119899DOby the usual

commutator

[119860 119861] = 119860119861minus119861119860

= sum

120574isinΓ+

119899

(1120574)(

120597120574

120597120585120574(119860) 120575

120574

(119861) minus120597120574

120597120585120574(119861) 120575

120574

(119860)) (67)

where 120597120574120597120585120574 and 120575120574 are determined by linearity and the rules

120597120574

120597120585120574(119886120585

120572

) = 120574 (120572

120574) 119886120585

120572minus120574

120575120574

(119886120585120572

) = 120575120574

(119886) 120585120572

(68)

32The Logarithm of a Symbol and the Logarithmic 2-CocycleWe define log 120585

119894for all 119894 = 1 119899 As in Section 2 the action

of log 120585119894on the algebraΨ

119899DO is via the commutator [log 120585

119894 sdot]

considered in (67)

[log 120585119894 119883] = sum

120574isinΓ+

119899

1120574

120597120574

120597120585120574(log 120585

119894) 120575

120574

(119883)

= sum

120574119894ge0

1120574119894

120597120574

119894

120597120585120574119894

119894

(log 120585119894) 120575

120574119894 (119883)

(69)

for all119883 isin ΨDO119899

Proposition 25 The linear operator [log 120585119894 sdot] defines a deriva-

tion of the (both associative and Lie) algebra Ψ119899119863119874 for all

119894 = 1 119899 The Lie algebra derivation [log 120585119894 sdot] is outer

Proof One verifies as in Section 2 that for any two symbols119860 and 119861 in ΨDO

[log 120585119894 119860119861] = [log 120585

119894 119860] 119861 +119860 [log 120585

119894 119861] (70)

so that [log 120585119894 sdot] is a derivation of the associative algebra

Equation (67) implies that [log 120585119894 sdot] is also a derivation of the

Lie algebra structure That [log 120585119894 sdot] is an outer derivation is

proven as in Proposition 13

Let 120591 be a 120575119894-invariant trace 119894 = 1 119899 so that 120591(120575

119894(119886)) =

0 for all 119886 isin A Then as in Section 2 we have

120591 (119886120575120574

(119887)) = (minus1)120574 120591 (119887120575120574 (119886)) (71)


Proposition 26 LetA be an algebra 120575119894derivations onA and

120591 a 120575119894-invariant trace 119894 = 1 119899 Then the linear functional

res Ψ119899119863119874 rarr CΨ

119899119863119874 rarr C defined by

res(sum120572isinΓ119899

119886120572120585120572

) = 120591 (119886minus120599) (72)

is a trace on Ψ119899119863119874

Proof It suffices to prove that for any 119886 119887 isin A and 120572 120573 isin Γ119899

res (119886120585120572119887120585120573) = res (119887120585120573119886120585120572) (73)

Let 120572 120573 isin Γ119899such that for some 120572

119894 120573

119895 we have 120572

119894 120573

119895ge 0

or 120572119894 120573

119895lt 0 then (73) holds by Proposition 3 Thus without

loss of generality we assume that for all 119894 120572119894ge 0 and 120573

119894lt 0

If 120572119894+ 120573

119894lt minus1 then using Proposition 3 we have (73) again

Now suppose that 120572119894+ 120573

119894ge minus1 and let 119896

119894= 120572

119894+ 120573

119894 We have

res (119886120585120572119887120585120573) = (120572

119896 + 120599) 120591 (119886120575

119896+120599

) (74)

On the other hand applying (12) 119899 times and using (71) weobtain

res (119887120585120573119886120585120572) = 120591((120573

119896 + 120599) 119887120575

119896+120599

(119886))

= (119896 minus 120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (minus1)119896+120599 (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

= (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

(75)

As in Section 21 we define the bilinear form ⟨119860 119861⟩ =

res(119860119861) for 119860 119861 isin Ψ119899DO We can prove that it is symmetric

and invariant and we will assume hereafter that it is nonde-generate Examples of nondegenerate bilinear forms as aboveappear in [22 23] for special choices of algebrasA

Theorem 27 The 2-cochains 119888119894(119860 119861) = res([log 120585

119894 119860]119861) 119894 =

1 119899 are Lie algebra 2-cocycles of Ψ119899119863119874

Proof Skew-symmetry is proven as in Theorem 14 Also asbefore a straightforward computation using that [log 120585

119894 sdot] is

a Lie algebra derivation yields 119888119894([119860 119861] 119862) + 119888

119894([119862 119860] 119861) +

119888119894([119861 119862] 119860) = 0

33Hierarchies of Centrally ExtendedAlgebras of Pseudodiffer-ential Symbols in Several Variables We present two examples

of hierarchies of subalgebras ofΨ119899DO and we prove that they

admit nontrivial central extensionsFor our first example we set 120572

119894= 119898 gt 0 for fixed 1 le 119894 le

119899 and

Ψ119899DO

120572119894

=

sum

120573isinΓ119899

119887120573120585120573

= sum

(1205731 120573119899)isinΓ119899

11988612057312058512057311 sdot sdot sdot 120585

120573119899

119899

isinΨ119899DO 120573

119894ge120572

119894

(76)

Theorem 28 The cocycle 119888119894defined by log 120585

119894is a nontrivial

cocycle for the subalgebra Ψ119899119863119874

120572119894

Hence it defines a nontriv-ial central extension of Ψ

119899119863119874

120572119894

Proof We identify Ψ119899DOlowast with Ψ

119899DO making use of the

nondegenerate bilinear form ⟨ ⟩ We also make the identi-fication Ψ

119899DOlowast

120572119894

≃ sum120579isinΓ119899

119897120579120585120579

120579119894le minus120572

119894minus 1 Let 119860 119861 isin

Ψ119899DO

120572119894

and assume that 119888119894is a coboundaryThen there exists

119871 isin Ψ119899DOlowast

120572119894

such that for 119860 119861 isin Ψ119899DO

120572119894

we have

119888 (119860 119861) = res ([119860 119861] 119871)

= res([

[

sum

120572isinΓ119899

119886120572120585120572

sum

120573isinΓ119899

119887120573120585120573]

]

sum

120579isinΓ119899

120579119894leminus120572119894minus1

119897120579120585120579

)

= res (sum((coefficient) 120575(120593) (119886120572) 120575

(])(119887120573) 120575

(120583)

(119897120579)

sdot 120585120572+120573+120579minus(]+120583+120593)

)) = 120591 (coefficientminus120599)

(77)

The coefficient of 120585minus120599 verifies that

120572119894+120573

119894+ 120579

119894+ (]

119894+120583

119894+120593

119894) = minus 1 (78)

on the 119894th position On the other hand we have that

119888 (119860 119861) = res ([log 120585119894 119860] 119861)

= res([log 120585119894 sum

120572isinΓ119899

119886120572120585120572

] sum

120573isinΓ119899

119887120573120585120573

)

= res (sum (coefficient) 120575(120595119894)119894

(119886120572) 120575

(120591119894)

(119887120573) 120585

120572+120573minus(120595119894+120591)

)


(79)

The coefficient of 120585minus120599 satisfies that

120572119894+120573

119894+ (120591

119894+120593

119894) = minus 1 (80)

on the 119894th position Comparing (78) with (80) for arbitrary119860 119861 we have that 120583

119894= 120579

119894 and this is a contradiction because

120579119894le minus120572

119894minus 1 and 120583

119894ge 0

For our second example let us say that 120572 ge 120573 if 120572119894ge 120573

119894

for all 119894 Fix 120572 isin Γ+119899 and define Ψ

119899DO

120572as the subalgebra of

Ψ119899DO with elements of the formsum

120573isinΓ119899

119887120573120585120573 such that 120573 ge 120572


Theorem 29 The cocycle generated by log 120585119894for each 119894 is a

nontrivial cocycle in Ψ119899119863119874

120572 Hence it defines a nontrivial

central extension of Ψ119899119863119874

120572

Proof Theproof is similar to the proof ofTheorem 28We seethat if the cocycle generated by log 120585

119894were trivial then (78)

and (80) would be true for all 119899 components and we wouldhave 120583

119894= 120579

119894for all 119894 which is impossible

34 The Dzhumadilrsquodaev ClassificationTheorem We sketch anew proof of the principal theorem of [23] It is a new proofin the sense that we argue by induction and we use toolsfrom homological algebra to perform the inductive step It isa sketch in the sense that full homological details are left forthe paper [24] We continue using the notation introduced inSection 31

Let 119875+119899

= sum120572120582120572119909120572

| 120572 isin Γ+

119899 be the algebra of

polynomials in variables 1199091 119909119899 and let 119875119899be the algebra

of Laurent power series of the form sum120572isinΓ119899

120582120572119909120572 such that the

number of 120572 isin Γ+

119899rsquos with nonzero 120582

120572is finite The action of

the derivation 120575119894on this algebra is determined by 120575

119894(119909

120572

) =

120572119894119909120572minus120598119894 with 120572 isin Γ

119899

LetU be another algebra with derivations 1205751 120575119899 Thetensor product U otimes 119875

119899becomes an associative algebra if we

endow it with the multiplication rule

(119906 otimes119891) (Votimes119892) = sum

120572isinΓ+

119899

1120572(119906120575

120572

(V) otimes 120575120572 (119891) 119892) (81)

Clearly this algebra contains U otimes 119875+

119899as a subalgebra and we

have the following

Proposition 30 There is an isomorphism between the asso-ciative algebra of formal pseudodifferential symbolsΨ

119899119863119874 and

U otimes 119875119899determined by the correspondence

Ψ119899119863119874 997888rarr Uotimes119875

119899

119906120585120572

997891997888rarr 119906otimes119909120572

(82)

This isomorphism determines a Lie algebra isomorphismbetween Ψ

119899119863119874 andU otimes 119875

119899

Now following [23] we letH119899be the Lie algebra associ-

ated toUotimes119875119899withU = 119875

119899Then identifying 119909120572

+119909120573

minus≃ 119909

120572

otimes119909120573

we see that an element ofH119899is of the form

sum

120572120573isinΓ119899

120582120572120573119909120572

+119909120573

minus (83)

and that the Lie bracket onH119899is given by

[119909120572

+119909120573

minus 119909

120572

+119909120573

minus] = sum

120574isinΓ+

119899

(1120574) (120597

minusand 120597

+) (119909

120572

+119909120573

minus 119909

120572

+119909120573

minus) (84)

where 120597+ 120597minusare derivations acting onH

119899as

120597120574

minus(119909

120572

+119909120573

minus) = 120574 (

120573

120574)119909

120572

+119909120573minus120574

minus

120597120574

+(119909

120572

+119909120573

minus) = 120574 (

120572

120574)119909

120572minus120574

+119909120573

minus

(85)

and 120597minusand 120597

+= 120597

minusotimes 120597

+minus 120597

+otimes 120597

minus

Definition 31 We define a linear function 1198910 on H119899via

1198910(119909minus120599

) = 1 and 1198910(119909120572

) = 0 if 120572 = minus120599

It is not hard to check that 1198910 is an outer derivation of theLie algebra H

119899 The following result is the main theorem of

this section

Theorem 32 The first group of cohomology of the Lie algebraH

119899with coefficients in H

119899 1198671

(H119899H

119899) is generated (as

space of outer derivations see Proposition 7) by 1198910 [log119909+119894 sdot]and [log119909

minus119894 sdot] 119894 = 1 119899

Proof We sketch an inductive proof of Theorem 32 Weconsider first the one variable case that is the basic elementsof the Lie algebra are 119909

+= 119909

+1 and 119909minus

= 119909minus1 We

writeH instead ofH1 Proposition 25 implies that [log119909+ sdot]

and [log119909minus sdot] are outer derivations and it is easy to check

that the derivations [log119909+ sdot] [log119909

minus sdot] and 1198910 are linearly

independent Now we have the following

Lemma 33 Let119863 be a derivation onHwith119863(119909plusmn) = 0Then

there is 120596 isinH such that

119863 = 120582 [log119909+ sdot] + 120573 [log119909

minus sdot] minus [120596 sdot] (86)

for some 120582 120573

Proof We know that [119906 1] = 0 and then [119863(1) 119906] = 0 forall 119906 isin H This implies that 119863(1) isin 119885(H) = C and thuswe can write 119863(1) = 120582 Now let 119863(119909

+) = sum119889

+

119899119898119909119899

+119909119898

minusand

119863(119909minus) = sum119889

minus

119899119898119909119899

+119909119898

minus We have

1 = [119909minus 119909

+] 997904rArr 120582 = [119863 (119909

minus) 119909

+] + [119909

minus 119863 (119909

+)]

997904rArr 120582 = sum119889minus

119899119898[119909

119899

+119909119898

minus 119909

+] +sum119889

+

119899119898[119909

minus 119909

119899

+119909119898

minus]

997904rArr 120582 = sum119889minus

119899119898119898119909

119899

+119909119898minus1minus

+sum119889+

119899119898119899119909

119899minus1+119909119898

minus

(87)

Equation (87) implies that 119889+119899119898

= 0 except when 119899 = 0 Thus1198890119898 = 0 and so 119863(119909

+) = sum119889

+

0119898119909119898

minus Analogously we have

that119863(119909minus) = sum119889

minus

1198990119909119899

+ Also (87) implies that

119889minus

119899119898119898119909

119899


= minus119889119899119898119899119909

119899minus1+119909119898

minus(88)

except when 119899 = minus1119898 = minus1 and (119899119898) = (0 0) We rewrite(88) as

(119898+ 1) 119889minus119899119898+1 = minus (119899 + 1) 119889119899+1119898 (89)


Then

119863(119909+) = sum

119898 =0minus1minus 119889

minus

0119898119909119898

minus+1198890minus1119909

minus1minus+119889

+

00 (90)

119863(119909minus) = sum

119899 =0minus1119889minus

1198990119909119899

++119889

minus10119909minus1++119889

minus

00 (91)

Now we write

119863(119909+) = 120582 [log119909

+ 119909

+] + 120573 [log119909

minus 119909

+] minus [120596 119909

+]

= 120573119909minus1minusminus [120596 119909

+]

(92)

for some 120596 to be determined Comparing (90) and (92) wehave that 120573 = 119889+0minus1 and

[120596 119909+] = sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120597minus(120596)

= sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120596

= sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

+119889+

00119909minus +

(93)

with [ 119909+] = 0 On the other hand

119863(119909minus) = 120582 [log119909

+ 119909

minus] + 120573 [log119909

minus 119909

minus] minus [120596 119909

minus]

= 120582119909minus1+minus [120596 119909

minus]

(94)

Comparing (91) and (94) and using (93) we have that 120582 = 119889minus01and

[120596 119909minus] = minus sum

119899 =0minus1119889minus

1198990119909119899

++minus119889

+

00 997904rArr 120597minus(120596)

= minus sum

119899 =0minus1119889minus

1198990119909119899

+minus119889

+

00 997904rArr 120596

= minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

minus119889minus

00119909+ +120596

(95)

with [120596 119909minus] = 0 Then we can take 120596 as

120596 = minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

+ sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

minus119889minus

00119909+ minus119889minus

00119909minus

(96)

Now we need the following result whose proof (see [24])is similar in spirit to the one we just carried out

Lemma 34 If 119863 is an outer derivation such that 119863(119909plusmn) = 0

then119863 is a scalar multiple of 1198910

Lemmas 33 and 34 imply that Theorem 32 holds in thecase 119899 = 1 In order to perform the inductive step we use anappropriate version of the Kunneth formula Again we referto [24]

We use Theorem 32 to classify central extensions of H119899

Theorem 1 of [23] tells us that H119899can be equipped with the

bilinear form ⟨119906 V⟩ = res(119906V) = (119906V)minus120599

and that this formis symmetric nondegenerate and invariant Then reasoningas in Theorem 27 we obtain that 119888

119894= res([log 120585

plusmn119894 119860]119861) and

1198880 = res([1198910 119860]119861) are Lie algebra cocycles of H119899 Now we

use the key Corollary 11 we have that ⟨1198910(119860) 119860⟩ = 0 for119860 = 119909

minus120599 and therefore 1198910 does not determine a centralextension On the other hand in [23 page 135] the authorshows that ⟨[log 120585

plusmn119894 119860] 119860⟩ = 0 for all 119860 We conclude that

the space of central extensions ofH119899is of dimension 2119899 The

case 119899 = 1 is discussed from a geometric point of view in [17Remark 416]

4 Applications

41 Manin Triples and Double Extensions of (Twisted) Pseu-dodifferential Symbols Manin triples are ubiquitous in inte-grable systems see for instance [17] They were introducedby Drinfelrsquod in his seminal paper [15] on Hopf algebras andthe quantum inverse scattering method

Definition 35 Three Lie algebras g gminus and g

+form a Manin

triple if the following conditions are satisfied

(1) The Lie algebras gminusand g

+are Lie subalgebras of g

such that g = gminusoplus g

minusas vector spaces

(2) There exists a nondegenerate invariant bilinear formon g such that g

+and g

minusare isotropic subspaces that

is the restrictions of this form to both g+and g

minus

vanish

For example if we assume that the bilinear form ⟨119860 119861⟩ =

res(119860119861) on ΨDO is nondegenerate then Proposition 6 tellsus that the algebras ΨDO DO and INT form a Manin triplewith respect to the bilinear form ⟨119860 119861⟩ In the same wayProposition 23 implies that the algebras ΨDO

120590 DO

120590 and

INT120590form aManin triple with respect to the twisted bilinear

form ⟨119860 119861⟩120590

Let us discuss the several variables case in some detailWe say that the order of the pseudodifferential operator 119871 =sum120572isinΓ119899

119886120572120585120572 is119873 if there is 120572 = (1205721 120572119899) with 120572119899 = 119873 such

that 119886120572119899

= 0 and for all 120573 = (1205731 120573119899) such that 120573119899gt 119873

we have 119886120573= 0 This definition is analogous to the notion of

order used by Parshin in [22]The Lie algebra Ψ

119899DO has two natural subalgebras

INT119899= 119871 isinΨ

119899DO | ord (119871) lt 0

DO119899= 119871 isinΨ

119899DO | ord (119871) ge 0

(97)

and clearly as a vector space Ψ119899DO is a direct sum of these

algebras Ψ119899DO = INT

119899oplus DO

119899

Proposition 36 If the bilinear form ⟨119860 119861⟩ = res(119860119861) is non-degenerate the algebras (Ψ

119899119863119874 119868119873119879

119899 119863119874

119899) form a Manin

triple with respect to ⟨ ⟩

Proof We reason as in the proof of Proposition 6


We can also define order in Ψ119899DO by using a fixed coor-

dinate 119886119894instead of 120572

119873 119894 = 119873 Then the Lie algebra Ψ

119899DO

has two natural subalgebras

INT119894=

(minus1)sum

(minusinfin)isinΓ119899

119886120572120585120572

| 119886120572isin119860

DO119894=

(119899)

sum

(0)isinΓ119899

119886120572120585120572

| 119886120572isin119860

(98)

As before we have that as a vector spaceΨ119899DO = INT

119894oplusDO

119894

Proposition 37 Thealgebras (Ψ119899119863119874 119868119873119879

119894 119863119874

119894) formaMa-

nin triple with respect to the bilinear form ⟨119860 119861⟩ = res(119860119861)

Proof This is again analogous to the proof of Proposition 6

Now we show how to construct further Manin tripleswith the help of central extensions using a general result byBordemann [25] which we now recall

LetA a nonassociative algebra over a field K and assumethat 119891 is a bilinear functional with the following properties

(1) 119891(119886119887 119888) = 119891(119886 119887119888) for all 119886 119887 119888 isin A(2) 119891(119886 119887) = 0 for all 119887 isin A implies 119886 = 0 and119891(119886 119887) = 0

for all 119886 isin A implies 119887 = 0

The pair (A 119891) is called a pseudometrized algebra or if 119891is symmetric a metrized algebra Bordemannrsquos result is thefollowing [25 Theorem 22]

Theorem 38 Let (A 119891) be a metrized algebra over a fieldK and let 119861 be a Lie algebra over K Suppose that there isa Lie homomorphism 120601 119861 rarr Der

119891(A) where Der

119891(A)

denotes the space of all 119891-antisymmetric derivations ofA (iethe derivations 119889 ofA for which 119891(119889119886 119886) +119891(119886 119889119886) = 0 for all119886 119886 isin A)

Let 119861lowast denote the dual space of 119861 We define119908 AtimesA rarr

Blowast as the bilinear antisymmetric map

(119886 119886) 997891997888rarr (119887 997891997888rarr119891 (120601 (119887) 119886 119886)) (99)

and for 119887 isin 119861 and 120573 isin 119861lowast we write 119887 sdot 120573 for the coadjoint

representation that is

(119887 sdot 120573 ()) = minus120573 (119887) (100)

We consider the vector space A119861= 119861 oplus 119860 oplus 119861

lowast and wedefine the following multiplication for 119887 isin 119861 119886 119886 isin A and120573 120573 isin 119861

lowast

(119887 + 119886 +120573) ( + 119886 +120573)

= 119887 + 120601 (119887) 119886 minus 120601 () 119886 + 119886119886 +119908 (119886 119886) + 119887 sdot 120573

minus sdot 120573

(101)

Moreover we define the following symmetric bilinear form 119891119861

onA119861

119891119861(119887 + 119886 +120573 + 119886 + 120573) = 120573 () + 120573 (119887) +119891 (119886 119886) (102)

Thepair (A119861 119891

119861) is ametrized algebra overK called the double

extension ofA by (120573 120595)

We sketch the proof of Theorem 38 in the case of interestfor us We assume that A is a Lie algebra and that 119891(119886 119887) =(119886 119887) is a bilineal symmetric formonAwhich is invariant andnondegenerate Then it is in fact easy to check that the bilin-eal form (102) is symmetric nondegenerate and invariant Onthe other hand that (101) defines a Lie bracket onA

119861follows

from a straightforward computation using the identities

119908(119886 [1198861015840

119886]) = 119908 (1198861015840

[119886 119886]) minus119908 (119886 [119886 1198861015840

])

119887 sdot 119908 (1198861015840

119886) = 119908 (1198861015840

120601 (119887) sdot 119886) +119908 (120601 (119887) sdot 1198861015840

119886)

(103)

for 119886 1198861015840 119886 isin A and 119887 isin 119861As a first application we considerA = ΨDO 119891(1198711 1198712) =

⟨1198711 1198712⟩ as in Section 2 and 119861 = 119887[log 120585 sdot] 119887 isin R 119861lowast = Rand we set 120601 119861 rarr Der

119891(A) as the inclusion map Then

119908(1198711 1198712) = ⟨[log 120585 1198711] 1198712⟩ and Theorem 38 gives us thedouble extension ΨDO = 119861 oplus ΨDO oplus 119861

lowast with bracket

[(1198871 1198711 1205731) (1198872 1198712 1205732)] = (0 1198871 [log 120585 1198712]

minus 1198872 [log 120585 1198711] + [1198711 1198712] ⟨[log 120585 1198711] 1198712⟩)(104)

and invariant symmetric linear form

⟨(1198871 1198711 1205731) (1198872 1198712 1205732)⟩

= 12057311198872 +12057321198871 + res (11987111198712) (105)

This extension is certainly known see [17] In an analogousway we obtain a double extension ΨDO

120590of the Lie algebra of

twisted pseudodifferential operatorsNow we note that the Lie algebra ΨDO (and also ΨDO

120590)

admits the (vector space) direct sum decomposition ΨDO =

DO oplus INT (resp ΨDO120590= DO

120590oplus INT

120590) in which

DO = 120572119888 +

119899

sum

119894=0119886119894120585119894

| 119886119894isinA 120572 isinR

INT = 120573 log 120585 +minus1sum

119894=minusinfin

119886119894120585119894

| 119886119894isinA 120573 isinR

(106)

and analogous definitions for the twisted case We have thefollowing

Corollary 39 If the bilinear form (102) is nondegenerate then

(Ψ119863119874119863119874 119868119873119879)

(Ψ120590119863119874119863119874

120590 119868119873119879

120590)

(107)

are Manin triples


Now we consider the doubly extended algebra Ψ119899DO of

Ψ119899DO that is the semidirect product of Ψ

119899DO = Ψ

119899DOoplusRsdot119888

119894

and the space of derivations 120573 log 120585119894| 120573 isin R We have the

vector space decomposition

Ψ119899DO = INT

119894oplus DO

119894 (108)

where

INT119894=

120573 log 120585119894+

(minus1)sum


119886120579120585120579

| 119886120579isin119860

DO119894=

120572119888119894+

(119899)

sum

(0)isinΓ119899

119886120579120585120579

| 119886120579isin119860

(109)

Theorem 38 yields the corollary

Corollary 40 The algebras (Ψ119899119863119874 119868119873119879

119894 119863119874

119894) form a Ma-

nin triple with respect to the bilinear form ⟨119860 + 1198871119888119894 +1205721 log 120585119894 119861 + 1198872119888119894 + 1205722 log 120585119894⟩ = res(119860119861) + 11988721205721 + 11988711205722 andLie bracket [(119860 1198871 1205721) (119861 1198872 1205722)] = ([119860 119861] + [1205722 log 120585119894 119860] minus[1205721 log 120585119894 119861] 119888119894(119860 119861) 0)

Theorem 38 also allows us to define a double extensionof Ψ

119899DO by considering the 119899-central extension R sdot 1198881 oplus

sdot sdot sdotR sdot 119888119899and the 119899 symbols log 120585

119894 We have the vector space

decomposition

Ψ119899DO = Ψ

119899DOoplus [(R sdot 1198881) oplus sdot sdot sdot oplus (R sdot 119888119899)]

oplus [(R sdot log 1205851) oplus sdot sdot sdot oplus (R sdot log 120585119899)] (110)

with Lie bracket given by

[(119860 120572 120573) (119861 120572 120573)] = ([119860 119861] + [120573 sdot log 120585 119860]

minus [120573 sdot log 120585 119861] 119888 (119860 119861) 0) (111)

and bilinear form

⟨119860 119861⟩ = res (119860 119861) + 120572 sdot 120573 +120573 sdot 120572 (112)

where 120572 120573 120572 and 120573 denote 119899-tuples log 120585 = (log 1205851 log 1205852 log 120585

119899) 0 = (0 0) isin R119899 and 119888(119860 119861) = (1198881(119860 119861)

119888119899(119860 119861))The operation ldquosdotrdquo is the usual dot product inR119899

for example 120573 sdot log 120585 = 1205731 log 120585 + sdot sdot sdot + 120573119899 log 120585119899

42 Hierarchies of Differential Equations In this last sectionwe apply ourwork to the construction of hierarchies of partialdifferential equations We begin with some standard factson hamiltonian systems modelled on (dual spaces of) Liealgebras (see for instance [17] or the recent paper [38])

Definition 41 If g is a Lie algebra and glowast is its dual space thefunctional derivative of a function 119865 glowast rarr R at 120583 isin glowast isthe unique element 120575119865120575120583 of g determined by

⟨]120575119865

120575120583⟩ =

119889

119889120598

10038161003816100381610038161003816100381610038161003816120598=0119865 (120583 + 120598]) (113)

for all ] isin glowast in which ⟨ ⟩ denotes a natural paring betweeng and glowast

Definition 42 The Lie-Poisson bracket on the dual space glowastis given by

119865 119866 (120583) = ⟨120583 [120575119865

120575120583120575119866

120575120583]⟩ (114)

in which 119865 119866 glowast rarr R and 120583 isin glowast

The corresponding equations of motion for a Hamilto-nian119867 glowast rarr R are

119889120583

119889119905= ⟨120583 [sdot

120575119867

120575120583]⟩ (115)

Definition 43 Let (g [ ]) be a Lie algebra and 119877 g rarr ga linear operator on g We define a bilinear antisymmetricbracket [ ]

119877on g by

[119875 119876]119877= [119877 (119875) 119876] + [119875 119877 (119876)] 119876 119875 isin g (116)

The operator 119877 is called a classical 119877-matrix if the bracket[ ]

119877satisfies the Jacobi identity

If g = gminusoplus g

+as vector space we choose 119877 = (12)(120587

+minus

120587minus) where 120587

plusmnare the projection operators on g

plusmn We write

119875plusmn= 120587

plusmn(119875) for119875 isin gThen [119875 119876]

119877= [119875

+ 119876

+]minus[119875

minus 119876

minus] and

119865 119866119877= ⟨120583 [

120575119865

120575120583120575119866

120575120583]

119877

⟩ (117)

If 119865 glowast rarr R satisfies the infinitesimal119860119889lowast-invariance con-dition

⟨120583 [120575119865

120575120583 sdot]⟩ = 0 (118)

for all 120583 isin glowast bracket (117) becomes

119865 119866119877= ⟨120583 [(

120575119865

120575120583)

+

120575119866

120575120583]⟩ (119)

It follows that if119867 is a hamiltonian function on glowast satisfying(118) the corresponding hamiltonian vector field is given by

119883119867(120583) sdot 119865 = ⟨120583 [minus(

120575119867

120575120583)

+

120575119865

120575120583]⟩ (120)

and if there exists a nondegenerate pairing ⟨ ⟩ between gand glowast such that the invariance condition

⟨119875 [119876 119877]⟩ = ⟨[119877 119875] 119876⟩ (121)

holds for 119875119876 119877 isin g then the corresponding equations ofmotion with respect to the Poisson structure (119) can bewritten as equations on g as

119889119875

119889119905= [(

120575119867

120575120583)

+

119875] (122)


421 The KP Hierarchy We apply the above remarks to theLie algebra ΨDO Let us recall that we have a bilinear sym-metric form on ΨDO which satisfies

⟨[119875 119876] 119878⟩ = ⟨[119878 119875] 119876⟩ 119878 119875 119876 isin ΨDO (123)

We assume that this form is nondegenerate The decompo-sition ΨDO = INT oplus DO = ΨDO

minusoplus ΨDO

+allows us to

define a new structure of Lie algebra [ ]119877on ΨDO using 119877-

matrices as aboveThe following proposition is standard seefor instance [5] or the recent work [38]

Proposition 44 Let 119867119896(119871) = res(119871119896) 119896 = 1 2 for 119871 isin

Ψ119863119874 Then

120575119867119896

120575119871= 119896119871

119896minus1 (124)

Moreover for any functional 119865 onΨ119863119874lowast 119867119896 119865(119871) = 0 and

119867119896 119867

119897119877(119871) = 0 for any 119877-matrix 119877

Equation (122) gives us the standard Kadomtsev-Petvi-ashvili hierarchy as a Hamiltonian equation

Theorem 45 Let 119867119896(119871) = res(119871119896) for 119871 isin Ψ119863119874 The cor-

responding Hamiltonian equation of motion with respect tosdot sdot

119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (125)

422 The Twisted KP Hierarchy We consider (59) andProposition 23 The Manin triple ΨDO

120590 DO

120590 and INT

120590

allows us to consider a new Lie bracket [ ]119877on ΨDO

120590as

in the previous subsection Proposition 44 is also valid in thetwisted context (see the proof appearing for instance in[38]) and therefore (122) yields the following result

Theorem 46 Let 119867119896(119871) = res(119871119896) for 119871 isin Ψ119863119874

120590 The

corresponding Hamiltonian equation of motion with respect tosdot sdot

119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (126)

A special case of (126) appears in [11] see (74) therein

423 The 119899-KP Hierarchy Now we consider Proposition 36As before the Manin triple Ψ

119899DO DO

119899 and INT

119899allows us

to induce a new Lie algebra structure on Ψ119899DO with bracket

[ ]119877 Once more we set 119867

119896(119871) = res(119871119896) for 119871 isin Ψ

119899DO

Equation (122) yields the equation of motion

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (127)

This is an 119899-variables generalization of KP of the kind con-sidered by Parshin in [22]

424 ZeroCurvature Equations Nowwe add a central exten-sion 119888 to ΨDO and we study ΨDO = ΨDO oplusR

Definition 47 Let (119860 120572) (119861 120573) isin ΨDO We define a bilinearform on ΨDO by

⟨(119860 120572) (119861 120573)⟩ = ⟨119860 119861⟩ + 120572120573 = res (119860119861) + 120572120573 (128)

We use this bilinear form to identify the dual space ofΨDO with itself

Proposition 48 Let 119867 (Ψ119863119874)lowast

rarr R be a Hamiltonianfunction The Hamiltonian equations of motion with respect tothe bracket (114) are

⟨120597119871

120597119905 119876⟩ = ⟨[

120597119867

120597119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (129)

119889119897

119889119905= 0 (130)

for any 119876 isin Ψ119863119874 in which ( ) = ⟨(119871 119897) (sdot sdot)⟩ isin (Ψ119863119874)lowast

Proof The Hamiltonian equations of motion on (ΨDO)lowastwith Hamiltonian119867 are

119889 ( )

119889119905sdot (119876 119903) = ⟨( ) [(119876 119903)

120575119867

120575 ( )]⟩

= ⟨( ) [(119876 119903) (120575119867

120575120575119867

120575)]⟩

= ⟨( ) ([119876120575119867

120575] 119888 (119876

120575119867

120575))⟩

(131)

If we set ( ) = ⟨(119871 119897) (sdot sdot)⟩ for some (119871 119897) isin ΨDO the lastequation becomes

⟨120597 (119871 119897)

120597119905 (119876 119903)⟩

= ⟨(119871 119897) ([119876120575119867

120575119871] 119888 (119876

120575119867

120575119871))⟩

= ⟨119871 [119876120575119867

120575119871]⟩+ 119897119888 (119876

120575119867

120575119871)

(132)

Using invariance of the bilinear form ⟨ ⟩ we write thisequation as

⟨120597119871

120597119905 119876⟩+ 119903

120597119897

120597119905= ⟨[

120575119867

120575119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (133)

Since this equation is valid for any (119876 119903) we conclude that119889119897119889119905 = 0 and we obtain (129)

In the context of current algebras (129) and (130) yield thezero curvature equations obtained by Reyman and Semenov-Tian-Shansky in [28]


Finally we restrict our attention to the subalgebra Vect =119886120585 119886 isin A In this case we can compute the cocycle 119888 rathereasily

res ([log 120585 119886120585] 119887120585) = res (119887120585 [log 120585 119886120585])

= res(119887120585infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 1205851minus119895)

= res(infin

sum

119895=0

(minus1)119895+1

119895(119887120575

119895

(119886) 1205852minus119895

+ 119887120575119895+1

(119886) 1205851minus119895))

=(minus1)3+1

3119887120575

3(119886) +

(minus1)2+1

2119887120575

3(119886)

= minus16120591 (119887120575

3(119886))

(134)

We consider the subalgebra Vect oplus R of ΨDO and weequip it with the bilinear form

⟨(119886120585 120572) (119887120585 120573)⟩ = 120591 (119886119887) + 120572120573 (135)

Then if we set (119871 119897) = (119906120585 119897) 119876 = 119887120585 and 120575119867120575119871 = ℎ(119906)120585and we use (129) and (130) which were derived without usingthe specific form of the inner product on ΨDO we obtain

⟨119889119906

119889119905120585 119887120585⟩ = ⟨[ℎ (119906) 120585 119906120585] 119887120585⟩

minus16119897120591 (119887120575

3(ℎ (119906)))

(136)

that is using that 120591 is nondegenerate

119889119906

119889119905= ℎ (119906) 120575 (119906) minus 119906120575 (ℎ (119906)) minus (

16) 119897120575

3(ℎ (119906)) (137)

an abstract form of the ubiquitous Korteweg-de Vries equa-tion

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Jarnishs Beltranrsquos research has been partially supported bythe Universidad del Desarrollo and the Project MECESUP-USACH PUC 0711 Enrique G Reyesrsquo research has beenpartially supported by the Project FONDECYT no 1111042and the USACH DICYT grant 041533RG

References

[1] V Ovsienko and C Roger ldquoDeforming the Lie algebra of vectorfields on 1198781 inside the Lie algebra of pseudodifferential symbolson 1198781rdquoTranslations of the AmericanMathematical Society Series2 vol 194 pp 211ndash226 1999

[2] L Guieu andC Roger LrsquoAlgebre et le Groupe de Virasoro Aspectsgeom Etriques et Algebriques Generalisations Les PublicationsCRM Montreal Canada 2007

[3] I M Gelfand and D B Fuchs ldquoCohomology of the Lie algebraof the vector field on the circlerdquo Functional Analysis and ItsApplications vol 2 pp 342ndash343 1969

[4] D Donin and B Khesin ldquoPseudodifferential symbols on Rie-mann surfaces and KricheverndashNovikov algebrasrdquo Communica-tions in Mathematical Physics vol 272 no 2 pp 507ndash527 2007

[5] L A Dickey Soliton Equations and Hamiltonian Systems vol12 of Advanced Series in Mathematical Physics World ScientificPublishing River Edge NJ USA 1991

[6] V G Kac and D H Peterson ldquoSpin and wedge representationsof infinite-dimensional Lie algebras and groupsrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 78 no 6 part 1 pp 3308ndash3312 1981

[7] M A Virasoro ldquoSubsidiary conditions and ghosts in dual-resonance modelsrdquo Physical Review D vol 1 no 10 pp 2933ndash2936 1970

[8] R Hernandez Heredero and E G Reyes ldquoGeometric integra-bility of the Camassa-Holm equation IIrdquo International Mathe-matics Research Notices vol 2012 no 13 pp 3089ndash3125 2012

[9] P Gorka and E G Reyes ldquoThe modified Hunter-Saxton equa-tionrdquo Journal of Geometry and Physics vol 62 no 8 pp 1793ndash1809 2012

[10] M Adler ldquoOn a trace functional for formal pseudo-differentialoperators and the symplectic structure of the Korteweg-devriestype equationsrdquo Inventiones Mathematicae vol 50 no 3 pp219ndash248 1978

[11] B Khesin V Lyubashenko and C Roger ldquoExtensions andcontractions of the Lie algebra of q-pseudodifferential symbolson the circlerdquo Journal of Functional Analysis vol 143 no 1 pp55ndash97 1997

[12] F Fathi-Zadeh andMKhalkhali ldquoThe algebra of formal twistedpseudodifferential symbols and a noncommutative residuerdquoLetters in Mathematical Physics vol 94 no 1 pp 41ndash61 2010

[13] A L Pirozerski and M A Semenov-Tian-Shansky ldquoGener-alized q-deformed Gelfand-Dickey structures on the groupof q-pseudo difference operatorsrdquo in Proceedings of the LDFaddeevrsquos Seminar on Mathematical Physics vol 201 of AMSTranslations Series 2 pp 211ndash238 American MathematicalSociety Providence RI USA 2000

[14] A L Pirozerski and M A Semenov-Tian-Shansky ldquoQ-pseu-dodifference universal Drinfeld-Sokolov reductionrdquo in Pro-ceedings of the St Petersburg Mathematical Society vol 7 ofSeries 2 AmericanMathematical Society 203 pp 169ndash199 AMSProvidence RI USA 2001

[15] V G Drinfelrsquod ldquoQuantum groupsrdquo in Proceedings of the ICMvol 1 pp 798ndash820 American Mathematical Society BerkeleyCalif USA 1986

[16] O S Kravchenko and B A Khesin ldquoA central extension of thealgebra of pseudodifferential symbolsrdquo Functional Analysis andIts Applications vol 25 no 2 pp 152ndash154 1991

[17] B Khesin and R Went The Geometry of Infinite-DimensionalGroups Springer Berlin Germany 2009

[18] B A Khesin ldquoAhierarchy of centrally extended algebras and thelogarithmof the derivative operatorrdquo InternationalMathematicsResearch Notices no 1 pp 1ndash5 1992

[19] E E Demidov ldquoOn the Kadomtsev-Petviashvili hierarchy witha noncommutative timespacerdquo Functional Analysis and ItsApplications vol 29 no 2 pp 131ndash133 1995


[20] E E Demidov ldquoNoncommutative deformation of the Kad-omtsev-Petviashvili hierarchyrdquo in Algebra 5 Vseross InstNauchn i Tekhn Inform (VINITI) Moscow Russia 1995(Russian) Journal of Mathematical Sciences vol 88 no 4 pp520ndash536 1998 (English)

[21] K Tenenblat Transformations of Manifolds and Applicationsto Differential Equations vol 93 of Pitman Monographs andSurveys in Pure and Applied Mathematics Longman HarlowUK 1998

[22] A N Parshin ldquoOn a ring of formal pseudo-differential opera-torsrdquo Proceedings of the Steklov Institute of Mathematics vol 1no 224 pp 266ndash280 1999

[23] A S Dzhumadilrsquodaev ldquoDerivations and central extensions ofthe Lie algebra of formal pseudodifferential operatorsrdquo Algebrai Analiz vol 6 no 1 pp 140ndash158 1994

[24] J Beltran andM Farinati ldquoTheHochschild and Lie (co)homol-ogy of the algebra of pseudodifferential operators in one andseveral variablesrdquo Preprint httpwwwcomplejidadsocialclwp-contentuploads201505HHPDiff-modpdf

[25] M Bordemann ldquoNondegenerate invariant bilinear forms onnonassociative algebrasrdquoActaMathematica Universitatis Come-nianae vol 66 no 2 pp 151ndash201 1997

[26] B A Khesin and I Zakharevich ldquoThe Lie-Poisson group ofpseudodifferential symbols and fractional KP-KdVhierarchiesrdquoComptes Rendus de lrsquoAcademie des Sciences Serie I vol 316 no6 pp 621ndash626 1993

[27] B Khesin and I Zakharevich ldquoPoisson-Lie group of pseudod-ifferential symbolsrdquo Communications in Mathematical Physicsvol 171 no 3 pp 475ndash530 1995

[28] AG Reyman andMA Semenov-Tian-Shansky ldquoCurrent alge-bras and nonlinear partial differential equationsrdquo Soviet Math-ematicsmdashDoklady vol 251 no 6 pp 1310ndash1314 1980

[29] M A Semenov-Tyan-Shanskii ldquoWhat is a classical r-matrixrdquoFunctional Analysis and Its Applications vol 17 no 4 pp 259ndash272 1983

[30] Y I Manin ldquoAlgebraic aspects of nonlinear differential equa-tionsrdquo Journal of Soviet Mathematics vol 11 pp 1ndash122 1979(Russian)

[31] M Mulase ldquoSolvability of the super KP equation and a general-ization of the Birkhoff decompositionrdquo Inventiones Mathemati-cae vol 92 no 1 pp 1ndash46 1988

[32] D B Fuks Cohomology of Infinite-Dimensional Lie AlgebrasContemporary Soviet Mathematics Consultants Bureau NewYork NY USA 1986

[33] A S Dzhumadilrsquodaev ldquoCentral extensions and invariant formsofCartan type Lie algebras of positive characteristicrdquo FunctionalAnalysis and Its Applications vol 18 no 4 pp 331ndash332 1984

[34] A S Dzhumadilrsquodaev ldquoCentral extensions of infinite-dimen-sional Lie algebrasrdquoFunctional Analysis and Its Applications vol26 no 4 pp 247ndash253 1992

[35] A I Kostrikin and I R Safarevic ldquoGraded Lie algebras of finitecharacteristicrdquoMathematics of the USSR-Izvestiya vol 3 no 2pp 237ndash304 1969

[36] W L Li ldquo2-cocyles on the algebra of differential operatorsrdquoJournal of Algebra vol 122 no 1 pp 64ndash80 1989

[37] M Wodzicki ldquoNoncommutative residue chapter I Fundamen-talsrdquo in K-Theory Arithmetic and Geometry Seminar MoscowUniversity 1984ndash1986 vol 1289 of Lecture Notes inMathematicspp 320ndash399 Springer Berlin Germany 1987

[38] A Eslami Rad and E G Reyes ldquoThe Kadomtsev-Petviashvilihierarchy and the Mulase factorization of formal Lie groupsrdquoJournal of Geometric Mechanics vol 5 no 3 pp 345ndash364 2013

Submit your manuscripts athttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Stochastic AnalysisInternational Journal of


In Section 2 we introduce the main objects considered inthis work We define formal pseudodifferential operators (orldquopseudodifferential symbolsrdquo) in one variable on an arbitraryassociative and commutative algebra and we construct theKravchenko-Khesin logarithmic 2-cocycle in this generalcontext We also construct a hierarchy of centrally extendedLie algebras of differential operators via the logarithmic 2-cocycle This construction generalizes a theorem by Khesin[18] on hierarchies of Lie algebras of differential operators onthe circle Finally motivated by [11 19 20] and the later paper[12] we consider twisted pseudodifferential symbols on arbi-trary associative and commutative algebras and in analogywith the untwisted case we construct central extensions andhierarchies of twisted centrally extended algebras

In Section 3 we consider pseudodifferential symbols inseveral independent variables on an arbitrary associativeand commutative algebra Our main motivation for con-sidering the several variables case in detail is the relativeabsence of examples of integrable equations in this contextIndeed besides the equations of the standard KP hierarchy[5] and their cousins there are not very many generalconstructions of integrable equations in several independentvariables Important exceptions are the equations introducedby Tenenblat and her coworkers (see [21] and referencestherein) and the hierarchies considered by Parshin [22]We construct central extensions in the several variables caseusing logarithmic cocycles and we also exhibit hierarchies ofLie algebras of differential operators in several independentvariables admitting central extensions We then considerthe work [23] by Dzhumadilrsquodaev in which he presents aclassification of central extensions The paper [23] is quitetechnical and it requires a very careful and critical readingand so we have decided to explain how to prove the mainresult of [23] using an inductive argument We present afull inductive proof of Dzhumadilrsquodaevrsquos theorem using sometechnical homological tools elsewhere see [24]

Finally in Section 4we introduceManin triples we definedouble extensions for the algebras of (twisted) pseudodiffer-ential symbols in one and several independent variables andusing a general algebraic theorem [25] we construct Manintriples for these algebras thereby putting [17 26ndash29] in avery general framework We also apply our algebraic resultsto the construction of integrable systems in one and severalindependent variables roughly following the techniques of[4 11 22]

2 The Algebra of Formal PseudodifferentialSymbols in One Variable

21 Basic Definitions and Preliminary Results Let A be anassociative and commutative algebra and let 120575 A rarr A

be a derivation on A that is 120575 is a linear map such that120575(119886119887) = 119886120575(119887) + 120575(119886)119887 for all 119886 119887 isin A The algebra of formaldifferential symbolsDO is generated byA and a symbol 120585withthe relation

120585 ∘ 119886 = 119886120585 + 120575 (119886) (1)

for all 119886 isin AThe algebraA is a subalgebra of DO and we canprove inductively that

120585119899

∘ 119886 =

119899

sum

119895=0(119899

119895) 120575

119895

(119886) 120585119899minus119895 (2)

for all 119886 isin A and 119899 ge 0We extend the algebra DO to obtain the algebra of pseu-

dodifferential symbols ΨDO by introducing differentiationswith negative exponents A general element of ΨDO is aformal series 119875 of the form

119875 =

119899

sum

minusinfin

119886119894120585119894 where 119886

119894isin A (3)

We set

120585minus1∘ 119886 =

infin

sum

119894=0(minus1)119894 120575119894 (119886) 120585minus1minus119894 (4)

and so (2) generalizes to

120585119899

∘ 119886 =

infin

sum

119895=0(119899

119895) 120575

119895

(119886) 120585119899minus119895 (5)

for all 119899 isin Z in which the binomial coefficient is defined by

(119899

119895) =

119899 (119899 minus 1) sdot sdot sdot (119899 minus 119895 + 1)119895

(6)

for 119899 119895 isin Z If 1198751 = sum119872

119894=minusinfin119886119894120585119894 and 1198752 = sum

119873

119895=minusinfin119887119895120585119895 then

1198751 ∘ 1198752 =infin

sum

119896=0

1119896

120597119896

120597120585119896(1198751) 120575

119896

(1198752) (7)

The Lie algebra structure on ΨDO is given by the usualcommutator [119860 119861] = 119860 ∘ 119861 minus 119861 ∘ 119860 so that for instance

[120585120572

119886120585119899

] =

infin

sum

119895=1(119899

119895) 120575

119895

(119886) 120585120572+119899minus119895

(8)

Lemma 1 For any nonnegative integer 119898 and 119886 119887 isin A wehave

120575119898

(119886119887) =

119898

sum

119895=0(119898

119895)120575

119898minus119895

(119886) 120575119895

(119887) (9)

Proof Proof is by induction

Let 120591 A rarr C be a 120575-invariant trace on A that is 120591 isa linear map satisfying 120591(119886119887) = 120591(119887119886) and 120591(120575(119886)) = 0 for all119886 119887 isin A For example if A = Diff(1198781) 120575 = 119889119889119909 the linearfunctional 120591 Diff(1198781) rarr C given by 120591(119891) = int

1198781 119891(119909)119889119909 is a

120575-invariant trace

Lemma 2 Let A be an algebra 120575 a derivation on A 120591 a 120575-invariant trace on A and 119898 a positive integer Then we canldquointegrate by partsrdquo that is for all 119886 119887 isin A we have

120591 (119887120575119898

(119886)) = (minus1)119898 120591 (119886120575119898 (119887)) (10)





defined by

res(119872

sum

119894=minusinfin

119886119894120585119894

) = 120591 (119886minus1) (11)



(119886

119887) = (minus1)119887 (

119887 minus 119886 minus 1119887

) (12)


res (119886120585119899 ∘ 119887120585119898) = res (119887120585119898 ∘ 119886120585119899) (13)


res (119886120585119899 ∘ 119887120585119898) =infin

sum

119895=0res((

119899

119895) 119886120575

119895

(119887) 120585119898+119899minus119895

)

= 120591((119899

119898 + 119899 + 1)119886120575

119898+119899+1(119887))

(14)


119898+119899+1 ) = 0 Similarly

res (119887120585119898 ∘ 119886120585119899) = 120591((119898

119898 + 119899 + 1)119886120575

119898+119899+1(119887)) = 0 (15)


res (119886120585119899 ∘ 119887120585119898) = res(infin

sum

119895=0(119899

119895) 119886120575

119895

(119887) 120585119898+119899minus119895

) (16)



res (119886120585119899 ∘ 119887120585119898) = res(infin

sum

119895=0(119899

119895) 119886120575

119895

(119887) 120585119898+119899minus119895

) = 0 (17)



res (119886120585119899 ∘ 119887120585119898) = 120591((119899

119896 + 1)119886120575

119896+1(119887))

= (119899

119896 + 1)120591 (119886120575

119896+1(119887))

(18)


res (119887120585119898 ∘ 119886120585119899) = 120591((119898

119896 + 1)119887120575

119896+1(119886))

= (119896 minus 119899

119896 + 1)120591 (119887120575

119896+1(119886))

= (minus1)119896+1 (119899

119896 + 1)120591 (119887120575

119896+1(119886))

= (minus1)119896+1 (minus1)119896+1 (119899

119896 + 1)120591 (119886120575

119896+1(119887))

= (119899

119896 + 1)120591 (119886120575

119896+1(119887))

(19)



convention

res (119860) = res(119899

sum

119894=minusinfin

119886119894(119909) 120597

119894

) = 120591 (119886minus1 (119909)) (20)



Now if 119875 = 119886119899120585119899

+ 119886119899minus1120585





DO =

119899

sum

119894=0119886119894120585119894

| 119886119894isinA

INT = minus1sum

119894=minusinfin

119886119894120585119894

| 119886119894isinA

(21)






119902 119862

119902


the formula

119889119888 (1198921 119892119902+1) = sum


sdot 119888 ([119892119904 119892

119905] 1198921 119892119904 119892119905 119892119902+1)

+ sum

1le119904le119902+1(minus1)119904 119892

119904119888 (1198921 119892119904 119892119902+1)

(22)







(g)






[(119883 119906) (119884 V)] = ([119883 119884] 119888 (119883 119884)) (23)




119888 ([119883 119884] 119885) + 119888 ([119885119883] 119884) + 119888 ([119884 119885] 119883) = 0 (24)




(gn)







120595 (119909) sdot 119909 = 0 (25)


1(g)







120595+[119886 sdot]


1198671(g) if and only if ⟨120575

120595(119909) 119909⟩ = 0 for all 119909 isin g



119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120585119905

= log 120585 (26)


119905=0 (119905

119895) = (minus1)119895+1119895 we obtain

[log 120585 119886120585119899] =infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 120585119899minus119895

(27)





119904(120573 minus 119904

119896) =

119904+119896

sum

119895=119904


119895(

120573

119904 + 119896 minus 119895)(

119895

119895 minus 119904) (28)








[log 120585 119886120585119899 ∘ 119887120585119898] = [log 120585 119886120585119899] ∘ 119887120585119898 + 119886120585119899

∘ [log 120585 119887120585119898] (29)


[log 120585 119886120585119899 ∘ 119887120585119898] = [log 120585infin

sum

119896=119900

(119899

119896)119886120575

119895

(119887) 120585119899+119898minus119896

]

=

infin

sum

119896=119900

infin

sum

119895=1(119899

119896)(minus1)119895+1

119895120575119895

(119886120575119896

(119887)) 120585119899+119898minus119896minus119895

(30)


[log 120585 119886120585119899] ∘ 119887120585119898 + 119886120585119899 ∘ [log 120585 119887120585119898]

= (

infin

sum

119895=119900

(minus1)119895+1

119895120575119895

(119886) 120585119899minus119895

)119887120585119898

+ 119886120585119899

(

infin

sum

119895=119900

(minus1)119895+1

119895120575119895

(119887) 120585119898minus119895

)

=

infin

sum

119895=1

infin

sum

119896=0(119899 minus 119895

119896)(minus1)119895+1

119895120575119895

(119886) 120575119896

(119887) 120585119899+119898minus119896minus119895

+

infin

sum

119895=1

infin

sum

119896=0(119899

119896)(minus1)119895+1

119895119886120575

119896

(120575119895

(119887)) 120585119899+119898minus119896minus119895

=

infin

sum

119895=1

infin

sum

119896=0(119899 minus 119895

119896)(minus1)119895+1

119895120575119895

(119886) 120575119896

(119887) 120585119899+119898minus119896minus119895

+

infin

sum

119895=1

infin

sum

119896=0(119899

119896)(minus1)119895+1

119895119886120575

119896+119895

(119887) 120585119899+119898minus119896minus119895

(31)


sum(119899

119896)(minus1)119895+1

119895120575119895

(119886120575119896

(119887)) (32)


sum

119895

sum

119894=119900

(119899

119896)(

119894

119895)(minus1)119895+1

119895120575119895minus119894

(119886) 120575119894+119896

(119887)

= sum

119895minus1

sum

119894=119900

(119899

119896)(

119894

119895)(minus1)119895+1

119895120575119895minus119894

(119886) 120575119894+119896

(119887)

+sum(119899

119896)(minus1)119895+1

119895119886120575

119903

(119887)

(33)


sum(119899 minus 119904

119897)(minus1)119904+1

119904120575119904

(119886) 120575119897

(119887)

+sum(119899

119897)(minus1)119904+1

119904119886120575

119903

(119887)

(34)


sum(119899

119896)(

119895

119894)(minus1)119895+1

119895= (

119899 minus 119904

119897)(minus1)119904+1

119904 (35)


119897+119904

sum

119895=119904

(119899

119897 + 119904 minus 119895)(

119895

119895 minus 119904)(minus1)119895+1

119895= (

119899 minus 119904

119897)(minus1)119904+1

119904 (36)



119888 (119860 119861) = res ([log 120585 119860] ∘ 119861) (37)



res ([log 120585 119886120585119899]) = 120591 ((minus1)119899

119899 + 1120575119899+1

(119886)) = 0 (38)


119888 ([119861 119860]) = res ([log 120585 119861]119860)

= res ([log 120585 119861119860] minus119861 [log 120585 119860])

= minus 119888 (119860 119861)

(39)




119888 (119891 (119909) 120597 119892 (119909) 120597) =16int119891

1015840

(119909) 11989210158401015840

(119909) 119889119909 (40)



119888 (119891120597119899

119892120597119898

) =119899119898

(119898 + 119899 + 1)int1198781119891(119898+1)

119892(119899)

119889120579 (41)


119888 (119860 119861) = int1198781([119909 119860] ∘ 119861) (42)



1015840


119872ΨDO =

119899

sum

119894=minusinfin

119886119894119863V

119894

| 119886119894isinM (43)


119863(sum

119899

119894=minusinfin119886119894119863119894

V) = 119886minus1119863



119875(res

119863(119860))Then as










119898≃ sum


119886119894120585119894



119894=119898119886119894120585119894 119861 =

sum119872

119894=119898119887119894120585119894



119897119894120585119894

isin ΨDOlowast

119898we

have

119888 (119860 119861) = res ([119860 119861] 119871) = res([

[

119872

sum

119894=119898

119886119894120585119894

119872

sum

119895=119898

119887119895120585119895]

]

sdot


119896=minusinfin

119897119896120585119896

)

= res( sum


119894)

sdot 120575(119905)

(119887119895) 120575

(119906)

(119897119896) 120585

119894+119895+119896minus(119905+119906+119904)

))


(44)


119894 + 119895 + 119896 minus (119905 + 119906 + 119904) = minus 1 (45)


119888 (119860 119861) = res ([log 120585 119860] 119861) = res([log 120585119872

sum

119894=119898

119886119894120585119894

]

sdot

119872

sum

119895=119898

119887119895120585119895

) = res( sum

119894119895119905119904=11989811989800(coefficient) 120575(119904) (119886

119894)

sdot 120575(119905)

(119887119895) 120585

119894+119895minus(119904+119905)


(46)


119894 + 119895 minus (119904 + 119905) = minus 1 (47)



119898









ΨDO120590=

119873

sum

119894=minusinfin

119886119894120585119894

119873 isinZ 119886119899isinA (48)


120585119886 = 120590 (119886) 120585 + 120575 (119886)

120585120585minus1= 120585

minus1120585 = 1

(49)


120585119899

119886 =

119899

sum

119894=0119875119894119899(120590 120575) (119886) 120585

119894

(50)





120585119899

119886 =

infin

sum


infin

sum


sdot 120590minus1(120575120590

minus1)119894119899


minus1)1198941(119886) 120585


(51)



119898=minusinfin119887119898120585119898

then

119860119861 =

119872

sum

119898=minusinfin

sum

119899lt0sum

119894ge0(minus1)|119894| 119886

119899120590minus1(120575120590



minus1)1198941(119887119898) 120585

119898+119899+|119894|

+

119872

sum

119898=minusinfin

119873

sum

119899=0

119899

sum

119895=0119886119899119875119895119899(120590 120575) (119887

119898) 120585

119898+119895

(52)




120591 (119887120590minus1(120575120590


minus1(120575120590

minus1)119894119898

(119886))




res(119899

sum

119894=minusinfin

119886119894120585119894

) = 120591 (119886minus1) (54)

is a trace on Ψ119863119874120590


120585119899

119886 =

infin

sum

119895=0(119899

119895) 120575

119895

(120590119899minus119895

(119886)) 120585119899minus119895

(55)


119905be a 1-


120585119905

119886 minus 120590119905(119886) 120585

119905

=

infin

sum

119895=1(119905

119895) 120575

119895

(120590119905minus119895) 120585

119905minus119895

(56)


[log 120585 119886] = 119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120590119905(119886) +

infin

sum

119895=1


119895120590minus119895

120575119895

(119886) 120585minus119895

(57)



rarr Ψ119863119874120590

defined by

[log 120585 119886120585119899] = 119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120590119905(119886) 120585

119899

+

infin

sum

119895=1


119895120590minus119895

120575119895

(119886) 120585minus119895

(58)

is a derivation




ΨDO120590= DO

120590oplus INT

120590 (59)

where

DO120590=

119899

sum

119894=0119886119894120585119894

| 119886119894isinA

INT120590=

minus1sum

119894=minusinfin

119886119894120585119894

| 119886119894isinA

(60)







algebras 119863119874120590and 119868119873119879



120590and 119868119873119879

120590

vanish








of ΨDO120590by

ΨDO120590119898

=

119872

sum

119894=119898

119886119894120585119894

| 119886119894isinA (61)




Γ119899= 120572 = (1205721 120572119894 120572119899) 120572119894 isinZ

Γ+

119899= 120572 isin Γ

119899 120572

119894isinZ

+

1le 119894 le 119899

120576119894= (0 0 1 0 0)

120599 =

119899

sum

119894=1120576119894isin Γ

119899

120572 =

119899

prod

119894=1120572119894 120572

119894isin Γ

+

119899

(62)


119899 120575120574(119887) = 12057512057411 ∘ sdot sdot sdot ∘ 120575

120574119899


1205751 120575119899 and

(120572

120574) = prod

119894

(120572119894

120574119894

) (63)


119894 with 119894 = 1 119899




120585119894119886 = 119886120585

119894+ 120575

119894(119886) (64)



119899


119886120585120572

sdot 119887120585120573

= sum

120574isinΓ+

119899

(120572

120574)119886120575

120574

(119887) 120585120572+120573minus120574

(65)


119899DOof formal


120585minus1119894119886 = sum

119895ge0(minus1)119895 120575119895

119894(119886) 120585


(66)


commutator

[119860 119861] = 119860119861minus119861119860

= sum

120574isinΓ+

119899

(1120574)(

120597120574

120597120585120574(119860) 120575

120574

(119861) minus120597120574

120597120585120574(119861) 120575

120574

(119860)) (67)


120597120574

120597120585120574(119886120585

120572

) = 120574 (120572

120574) 119886120585

120572minus120574

120575120574

(119886120585120572

) = 120575120574

(119886) 120585120572

(68)





119894 sdot]

considered in (67)

[log 120585119894 119883] = sum

120574isinΓ+

119899

1120574

120597120574

120597120585120574(log 120585

119894) 120575

120574

(119883)

= sum

120574119894ge0

1120574119894

120597120574

119894

120597120585120574119894

119894

(log 120585119894) 120575

120574119894 (119883)

(69)






[log 120585119894 119860119861] = [log 120585

119894 119860] 119861 +119860 [log 120585

119894 119861] (70)






119894(119886)) =


120591 (119886120575120574

(119887)) = (minus1)120574 120591 (119887120575120574 (119886)) (71)




res Ψ119899119863119874 rarr CΨ

119899119863119874 rarr C defined by


119886120572120585120572

) = 120591 (119886minus120599) (72)

is a trace on Ψ119899119863119874


res (119886120585120572119887120585120573) = res (119887120585120573119886120585120572) (73)


119894 120573

119895 we have 120572

119894 120573

119895ge 0

or 120572119894 120573



119894lt 0

If 120572119894+ 120573


Now suppose that 120572119894+ 120573


119894= 120572

119894+ 120573

119894 We have

res (119886120585120572119887120585120573) = (120572

119896 + 120599) 120591 (119886120575

119896+120599

) (74)


res (119887120585120573119886120585120572) = 120591((120573

119896 + 120599) 119887120575

119896+120599

(119886))

= (119896 minus 120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (minus1)119896+120599 (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

= (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

(75)





119894 119860]119861) 119894 =



119894 sdot] is


119894([119862 119860] 119861) +

119888119894([119861 119862] 119860) = 0




119894= 119898 gt 0 for fixed 1 le 119894 le

119899 and

Ψ119899DO

120572119894

=

sum

120573isinΓ119899

119887120573120585120573

= sum

(1205731 120573119899)isinΓ119899

11988612057312058512057311 sdot sdot sdot 120585

120573119899

119899

isinΨ119899DO 120573

119894ge120572

119894

(76)




120572119894


119899119863119874

120572119894




119899DOlowast

120572119894

≃ sum120579isinΓ119899

119897120579120585120579

120579119894le minus120572

119894minus 1 Let 119860 119861 isin

Ψ119899DO

120572119894



120572119894


120572119894

we have

119888 (119860 119861) = res ([119860 119861] 119871)

= res([

[

sum

120572isinΓ119899

119886120572120585120572

sum

120573isinΓ119899

119887120573120585120573]

]

sum

120579isinΓ119899

120579119894leminus120572119894minus1

119897120579120585120579

)


(])(119887120573) 120575

(120583)

(119897120579)

sdot 120585120572+120573+120579minus(]+120583+120593)


(77)


120572119894+120573

119894+ 120579

119894+ (]

119894+120583

119894+120593

119894) = minus 1 (78)


119888 (119860 119861) = res ([log 120585119894 119860] 119861)

= res([log 120585119894 sum

120572isinΓ119899

119886120572120585120572

] sum

120573isinΓ119899

119887120573120585120573

)


(119886120572) 120575

(120591119894)

(119887120573) 120585

120572+120573minus(120595119894+120591)

)


(79)


120572119894+120573

119894+ (120591

119894+120593

119894) = minus 1 (80)


119894= 120579


120579119894le minus120572

119894minus 1 and 120583

119894ge 0


119894


119899DO



120573isinΓ119899

119887120573120585120573 such that 120573 ge 120572






120572




119894= 120579



Let 119875+119899

= sum120572120582120572119909120572

| 120572 isin Γ+




120582120572119909120572 such that the





119894(119909

120572

) =

120572119894119909120572minus120598119894 with 120572 isin Γ

119899





120572isinΓ+

119899

1120572(119906120575

120572

(V) otimes 120575120572 (119891) 119892) (81)



have the following


119899119863119874 and


Ψ119899119863119874 997888rarr Uotimes119875

119899

119906120585120572

997891997888rarr 119906otimes119909120572

(82)


119899119863119874 andU otimes 119875

119899




+119909120573

minus≃ 119909

120572

otimes119909120573


sum

120572120573isinΓ119899

120582120572120573119909120572

+119909120573

minus (83)


[119909120572

+119909120573

minus 119909

120572

+119909120573

minus] = sum

120574isinΓ+

119899

(1120574) (120597

minusand 120597

+) (119909

120572

+119909120573

minus 119909

120572

+119909120573

minus) (84)


119899as

120597120574

minus(119909

120572

+119909120573

minus) = 120574 (

120573

120574)119909

120572

+119909120573minus120574

minus

120597120574

+(119909

120572

+119909120573

minus) = 120574 (

120572

120574)119909

120572minus120574

+119909120573

minus

(85)


+= 120597

minusotimes 120597

+minus 120597

+otimes 120597

minus


1198910(119909minus120599

) = 1 and 1198910(119909120572

) = 0 if 120572 = minus120599



this section



119899 1198671

(H119899H



minus119894 sdot] 119894 = 1 119899


+= 119909

+1 and 119909minus

= 119909minus1 We








119863 = 120582 [log119909+ sdot] + 120573 [log119909


for some 120582 120573


+) = sum119889

+

119899119898119909119899

+119909119898

minusand

119863(119909minus) = sum119889

minus

119899119898119909119899

+119909119898

minus We have

1 = [119909minus 119909

+] 997904rArr 120582 = [119863 (119909

minus) 119909

+] + [119909

minus 119863 (119909

+)]

997904rArr 120582 = sum119889minus

119899119898[119909

119899

+119909119898

minus 119909

+] +sum119889

+

119899119898[119909

minus 119909

119899

+119909119898

minus]

997904rArr 120582 = sum119889minus

119899119898119898119909

119899


+sum119889+

119899119898119899119909

119899minus1+119909119898

minus

(87)



+) = sum119889

+

0119898119909119898


that119863(119909minus) = sum119889

minus

1198990119909119899


119889minus

119899119898119898119909

119899


= minus119889119899119898119899119909

119899minus1+119909119898

minus(88)


(119898+ 1) 119889minus119899119898+1 = minus (119899 + 1) 119889119899+1119898 (89)


Then

119863(119909+) = sum

119898 =0minus1minus 119889

minus

0119898119909119898


minus1minus+119889

+

00 (90)

119863(119909minus) = sum

119899 =0minus1119889minus

1198990119909119899

++119889

minus10119909minus1++119889

minus

00 (91)

Now we write

119863(119909+) = 120582 [log119909

+ 119909

+] + 120573 [log119909

minus 119909

+] minus [120596 119909

+]


+]

(92)


[120596 119909+] = sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120597minus(120596)

= sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120596

= sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

+119889+

00119909minus +

(93)


119863(119909minus) = 120582 [log119909

+ 119909

minus] + 120573 [log119909

minus 119909

minus] minus [120596 119909

minus]

= 120582119909minus1+minus [120596 119909

minus]

(94)



119899 =0minus1119889minus

1198990119909119899

++minus119889

+

00 997904rArr 120597minus(120596)

= minus sum

119899 =0minus1119889minus

1198990119909119899

+minus119889

+

00 997904rArr 120596

= minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

minus119889minus

00119909+ +120596

(95)


120596 = minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

+ sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

minus119889minus

00119909+ minus119889minus

00119909minus

(96)









119894= res([log 120585

plusmn119894 119860]119861) and







4 Applications



+form a Manin







+and g



minus

vanish



120590 DO

120590 and


form ⟨119860 119861⟩120590



that 119886120572119899





INT119899= 119871 isinΨ

119899DO | ord (119871) lt 0

DO119899= 119871 isinΨ

119899DO | ord (119871) ge 0

(97)



119899oplus DO

119899


119899119863119874 119868119873119879

119899 119863119874








119899DO


INT119894=

(minus1)sum


119886120572120585120572

| 119886120572isin119860

DO119894=

(119899)

sum

(0)isinΓ119899

119886120572120585120572

| 119886120572isin119860

(98)


119894oplusDO

119894


119894 119863119874

119894) formaMa-









119891(A) where Der

119891(A)




(119886 119886) 997891997888rarr (119887 997891997888rarr119891 (120601 (119887) 119886 119886)) (99)



(119887 sdot 120573 ()) = minus120573 (119887) (100)



lowast

(119887 + 119886 +120573) ( + 119886 +120573)

= 119887 + 120601 (119887) 119886 minus 120601 () 119886 + 119886119886 +119908 (119886 119886) + 119887 sdot 120573

minus sdot 120573

(101)


onA119861

119891119861(119887 + 119886 +120573 + 119886 + 120573) = 120573 () + 120573 (119887) +119891 (119886 119886) (102)

Thepair (A119861 119891




119861follows


119908(119886 [1198861015840

119886]) = 119908 (1198861015840

[119886 119886]) minus119908 (119886 [119886 1198861015840

])

119887 sdot 119908 (1198861015840

119886) = 119908 (1198861015840

120601 (119887) sdot 119886) +119908 (120601 (119887) sdot 1198861015840

119886)

(103)





lowast with bracket

[(1198871 1198711 1205731) (1198872 1198712 1205732)] = (0 1198871 [log 120585 1198712]

minus 1198872 [log 120585 1198711] + [1198711 1198712] ⟨[log 120585 1198711] 1198712⟩)(104)


⟨(1198871 1198711 1205731) (1198872 1198712 1205732)⟩

= 12057311198872 +12057321198871 + res (11987111198712) (105)




120590)



120590oplus INT

120590) in which

DO = 120572119888 +

119899

sum

119894=0119886119894120585119894

| 119886119894isinA 120572 isinR


119894=minusinfin

119886119894120585119894

| 119886119894isinA 120573 isinR

(106)



(Ψ119863119874119863119874 119868119873119879)

(Ψ120590119863119874119863119874

120590 119868119873119879

120590)

(107)

are Manin triples




119899DO = Ψ


119894



Ψ119899DO = INT

119894oplus DO

119894 (108)

where

INT119894=

120573 log 120585119894+

(minus1)sum


119886120579120585120579

| 119886120579isin119860

DO119894=

120572119888119894+

(119899)

sum

(0)isinΓ119899

119886120579120585120579

| 119886120579isin119860

(109)



119894 119863119874

119894) form a Ma-






decomposition

Ψ119899DO = Ψ




[(119860 120572 120573) (119861 120572 120573)] = ([119860 119861] + [120573 sdot log 120585 119860]

minus [120573 sdot log 120585 119861] 119888 (119860 119861) 0) (111)

and bilinear form

⟨119860 119861⟩ = res (119860 119861) + 120572 sdot 120573 +120573 sdot 120572 (112)


119899) 0 = (0 0) isin R119899 and 119888(119860 119861) = (1198881(119860 119861)





⟨]120575119865

120575120583⟩ =

119889

119889120598

10038161003816100381610038161003816100381610038161003816120598=0119865 (120583 + 120598]) (113)



119865 119866 (120583) = ⟨120583 [120575119865

120575120583120575119866

120575120583]⟩ (114)



119889120583

119889119905= ⟨120583 [sdot

120575119867

120575120583]⟩ (115)


119877on g by

[119875 119876]119877= [119877 (119875) 119876] + [119875 119877 (119876)] 119876 119875 isin g (116)





+minus



plusmn We write

119875plusmn= 120587


119877= [119875

+ 119876

+]minus[119875

minus 119876

minus] and

119865 119866119877= ⟨120583 [

120575119865

120575120583120575119866

120575120583]

119877

⟩ (117)


⟨120583 [120575119865

120575120583 sdot]⟩ = 0 (118)


119865 119866119877= ⟨120583 [(

120575119865

120575120583)

+

120575119866

120575120583]⟩ (119)


119883119867(120583) sdot 119865 = ⟨120583 [minus(

120575119867

120575120583)

+

120575119865

120575120583]⟩ (120)


⟨119875 [119876 119877]⟩ = ⟨[119877 119875] 119876⟩ (121)


119889119875

119889119905= [(

120575119867

120575120583)

+

119875] (122)



⟨[119875 119876] 119878⟩ = ⟨[119878 119875] 119876⟩ 119878 119875 119876 isin ΨDO (123)


minusoplus ΨDO

+allows us to




Ψ119863119874 Then

120575119867119896

120575119871= 119896119871

119896minus1 (124)


119867119896 119867

119897119877(119871) = 0 for any 119877-matrix 119877




119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (125)


120590 DO

120590 and INT

120590


120590as



120590 The


119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (126)



119899DO DO

119899 and INT

119899allows us


[ ]119877 Once more we set 119867

119896(119871) = res(119871119896) for 119871 isin Ψ

119899DO


120597119871

120597119905= [(119896119871

119896minus1)+

119871] (127)




⟨(119860 120572) (119861 120573)⟩ = ⟨119860 119861⟩ + 120572120573 = res (119860119861) + 120572120573 (128)




⟨120597119871

120597119905 119876⟩ = ⟨[

120597119867

120597119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (129)

119889119897

119889119905= 0 (130)



119889 ( )

119889119905sdot (119876 119903) = ⟨( ) [(119876 119903)

120575119867

120575 ( )]⟩

= ⟨( ) [(119876 119903) (120575119867

120575120575119867

120575)]⟩

= ⟨( ) ([119876120575119867

120575] 119888 (119876

120575119867

120575))⟩

(131)


⟨120597 (119871 119897)

120597119905 (119876 119903)⟩

= ⟨(119871 119897) ([119876120575119867

120575119871] 119888 (119876

120575119867

120575119871))⟩

= ⟨119871 [119876120575119867

120575119871]⟩+ 119897119888 (119876

120575119867

120575119871)

(132)


⟨120597119871

120597119905 119876⟩+ 119903

120597119897

120597119905= ⟨[

120575119867

120575119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (133)





res ([log 120585 119886120585] 119887120585) = res (119887120585 [log 120585 119886120585])

= res(119887120585infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 1205851minus119895)

= res(infin

sum

119895=0

(minus1)119895+1

119895(119887120575

119895

(119886) 1205852minus119895

+ 119887120575119895+1

(119886) 1205851minus119895))

=(minus1)3+1

3119887120575

3(119886) +

(minus1)2+1

2119887120575

3(119886)

= minus16120591 (119887120575

3(119886))

(134)


⟨(119886120585 120572) (119887120585 120573)⟩ = 120591 (119886119887) + 120572120573 (135)


⟨119889119906

119889119905120585 119887120585⟩ = ⟨[ℎ (119906) 120585 119906120585] 119887120585⟩

minus16119897120591 (119887120575

3(ℎ (119906)))

(136)


119889119906

119889119905= ℎ (119906) 120575 (119906) minus 119906120575 (ℎ (119906)) minus (

16) 119897120575

3(ℎ (119906)) (137)




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













defined by

res(119872

sum

119894=minusinfin

119886119894120585119894

) = 120591 (119886minus1) (11)



(119886

119887) = (minus1)119887 (

119887 minus 119886 minus 1119887

) (12)


res (119886120585119899 ∘ 119887120585119898) = res (119887120585119898 ∘ 119886120585119899) (13)


res (119886120585119899 ∘ 119887120585119898) =infin

sum

119895=0res((

119899

119895) 119886120575

119895

(119887) 120585119898+119899minus119895

)

= 120591((119899

119898 + 119899 + 1)119886120575

119898+119899+1(119887))

(14)


119898+119899+1 ) = 0 Similarly

res (119887120585119898 ∘ 119886120585119899) = 120591((119898

119898 + 119899 + 1)119886120575

119898+119899+1(119887)) = 0 (15)


res (119886120585119899 ∘ 119887120585119898) = res(infin

sum

119895=0(119899

119895) 119886120575

119895

(119887) 120585119898+119899minus119895

) (16)



res (119886120585119899 ∘ 119887120585119898) = res(infin

sum

119895=0(119899

119895) 119886120575

119895

(119887) 120585119898+119899minus119895

) = 0 (17)



res (119886120585119899 ∘ 119887120585119898) = 120591((119899

119896 + 1)119886120575

119896+1(119887))

= (119899

119896 + 1)120591 (119886120575

119896+1(119887))

(18)


res (119887120585119898 ∘ 119886120585119899) = 120591((119898

119896 + 1)119887120575

119896+1(119886))

= (119896 minus 119899

119896 + 1)120591 (119887120575

119896+1(119886))

= (minus1)119896+1 (119899

119896 + 1)120591 (119887120575

119896+1(119886))

= (minus1)119896+1 (minus1)119896+1 (119899

119896 + 1)120591 (119886120575

119896+1(119887))

= (119899

119896 + 1)120591 (119886120575

119896+1(119887))

(19)



convention

res (119860) = res(119899

sum

119894=minusinfin

119886119894(119909) 120597

119894

) = 120591 (119886minus1 (119909)) (20)



Now if 119875 = 119886119899120585119899

+ 119886119899minus1120585





DO =

119899

sum

119894=0119886119894120585119894

| 119886119894isinA

INT = minus1sum

119894=minusinfin

119886119894120585119894

| 119886119894isinA

(21)






119902 119862

119902


the formula

119889119888 (1198921 119892119902+1) = sum


sdot 119888 ([119892119904 119892

119905] 1198921 119892119904 119892119905 119892119902+1)

+ sum

1le119904le119902+1(minus1)119904 119892

119904119888 (1198921 119892119904 119892119902+1)

(22)







(g)






[(119883 119906) (119884 V)] = ([119883 119884] 119888 (119883 119884)) (23)




119888 ([119883 119884] 119885) + 119888 ([119885119883] 119884) + 119888 ([119884 119885] 119883) = 0 (24)




(gn)







120595 (119909) sdot 119909 = 0 (25)


1(g)







120595+[119886 sdot]


1198671(g) if and only if ⟨120575

120595(119909) 119909⟩ = 0 for all 119909 isin g



119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120585119905

= log 120585 (26)


119905=0 (119905

119895) = (minus1)119895+1119895 we obtain

[log 120585 119886120585119899] =infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 120585119899minus119895

(27)





119904(120573 minus 119904

119896) =

119904+119896

sum

119895=119904


119895(

120573

119904 + 119896 minus 119895)(

119895

119895 minus 119904) (28)








[log 120585 119886120585119899 ∘ 119887120585119898] = [log 120585 119886120585119899] ∘ 119887120585119898 + 119886120585119899

∘ [log 120585 119887120585119898] (29)


[log 120585 119886120585119899 ∘ 119887120585119898] = [log 120585infin

sum

119896=119900

(119899

119896)119886120575

119895

(119887) 120585119899+119898minus119896

]

=

infin

sum

119896=119900

infin

sum

119895=1(119899

119896)(minus1)119895+1

119895120575119895

(119886120575119896

(119887)) 120585119899+119898minus119896minus119895

(30)


[log 120585 119886120585119899] ∘ 119887120585119898 + 119886120585119899 ∘ [log 120585 119887120585119898]

= (

infin

sum

119895=119900

(minus1)119895+1

119895120575119895

(119886) 120585119899minus119895

)119887120585119898

+ 119886120585119899

(

infin

sum

119895=119900

(minus1)119895+1

119895120575119895

(119887) 120585119898minus119895

)

=

infin

sum

119895=1

infin

sum

119896=0(119899 minus 119895

119896)(minus1)119895+1

119895120575119895

(119886) 120575119896

(119887) 120585119899+119898minus119896minus119895

+

infin

sum

119895=1

infin

sum

119896=0(119899

119896)(minus1)119895+1

119895119886120575

119896

(120575119895

(119887)) 120585119899+119898minus119896minus119895

=

infin

sum

119895=1

infin

sum

119896=0(119899 minus 119895

119896)(minus1)119895+1

119895120575119895

(119886) 120575119896

(119887) 120585119899+119898minus119896minus119895

+

infin

sum

119895=1

infin

sum

119896=0(119899

119896)(minus1)119895+1

119895119886120575

119896+119895

(119887) 120585119899+119898minus119896minus119895

(31)


sum(119899

119896)(minus1)119895+1

119895120575119895

(119886120575119896

(119887)) (32)


sum

119895

sum

119894=119900

(119899

119896)(

119894

119895)(minus1)119895+1

119895120575119895minus119894

(119886) 120575119894+119896

(119887)

= sum

119895minus1

sum

119894=119900

(119899

119896)(

119894

119895)(minus1)119895+1

119895120575119895minus119894

(119886) 120575119894+119896

(119887)

+sum(119899

119896)(minus1)119895+1

119895119886120575

119903

(119887)

(33)


sum(119899 minus 119904

119897)(minus1)119904+1

119904120575119904

(119886) 120575119897

(119887)

+sum(119899

119897)(minus1)119904+1

119904119886120575

119903

(119887)

(34)


sum(119899

119896)(

119895

119894)(minus1)119895+1

119895= (

119899 minus 119904

119897)(minus1)119904+1

119904 (35)


119897+119904

sum

119895=119904

(119899

119897 + 119904 minus 119895)(

119895

119895 minus 119904)(minus1)119895+1

119895= (

119899 minus 119904

119897)(minus1)119904+1

119904 (36)



119888 (119860 119861) = res ([log 120585 119860] ∘ 119861) (37)



res ([log 120585 119886120585119899]) = 120591 ((minus1)119899

119899 + 1120575119899+1

(119886)) = 0 (38)


119888 ([119861 119860]) = res ([log 120585 119861]119860)

= res ([log 120585 119861119860] minus119861 [log 120585 119860])

= minus 119888 (119860 119861)

(39)




119888 (119891 (119909) 120597 119892 (119909) 120597) =16int119891

1015840

(119909) 11989210158401015840

(119909) 119889119909 (40)



119888 (119891120597119899

119892120597119898

) =119899119898

(119898 + 119899 + 1)int1198781119891(119898+1)

119892(119899)

119889120579 (41)


119888 (119860 119861) = int1198781([119909 119860] ∘ 119861) (42)



1015840


119872ΨDO =

119899

sum

119894=minusinfin

119886119894119863V

119894

| 119886119894isinM (43)


119863(sum

119899

119894=minusinfin119886119894119863119894

V) = 119886minus1119863



119875(res

119863(119860))Then as










119898≃ sum


119886119894120585119894



119894=119898119886119894120585119894 119861 =

sum119872

119894=119898119887119894120585119894



119897119894120585119894

isin ΨDOlowast

119898we

have

119888 (119860 119861) = res ([119860 119861] 119871) = res([

[

119872

sum

119894=119898

119886119894120585119894

119872

sum

119895=119898

119887119895120585119895]

]

sdot


119896=minusinfin

119897119896120585119896

)

= res( sum


119894)

sdot 120575(119905)

(119887119895) 120575

(119906)

(119897119896) 120585

119894+119895+119896minus(119905+119906+119904)

))


(44)


119894 + 119895 + 119896 minus (119905 + 119906 + 119904) = minus 1 (45)


119888 (119860 119861) = res ([log 120585 119860] 119861) = res([log 120585119872

sum

119894=119898

119886119894120585119894

]

sdot

119872

sum

119895=119898

119887119895120585119895

) = res( sum

119894119895119905119904=11989811989800(coefficient) 120575(119904) (119886

119894)

sdot 120575(119905)

(119887119895) 120585

119894+119895minus(119904+119905)


(46)


119894 + 119895 minus (119904 + 119905) = minus 1 (47)



119898









ΨDO120590=

119873

sum

119894=minusinfin

119886119894120585119894

119873 isinZ 119886119899isinA (48)


120585119886 = 120590 (119886) 120585 + 120575 (119886)

120585120585minus1= 120585

minus1120585 = 1

(49)


120585119899

119886 =

119899

sum

119894=0119875119894119899(120590 120575) (119886) 120585

119894

(50)





120585119899

119886 =

infin

sum


infin

sum


sdot 120590minus1(120575120590

minus1)119894119899


minus1)1198941(119886) 120585


(51)



119898=minusinfin119887119898120585119898

then

119860119861 =

119872

sum

119898=minusinfin

sum

119899lt0sum

119894ge0(minus1)|119894| 119886

119899120590minus1(120575120590



minus1)1198941(119887119898) 120585

119898+119899+|119894|

+

119872

sum

119898=minusinfin

119873

sum

119899=0

119899

sum

119895=0119886119899119875119895119899(120590 120575) (119887

119898) 120585

119898+119895

(52)




120591 (119887120590minus1(120575120590


minus1(120575120590

minus1)119894119898

(119886))




res(119899

sum

119894=minusinfin

119886119894120585119894

) = 120591 (119886minus1) (54)

is a trace on Ψ119863119874120590


120585119899

119886 =

infin

sum

119895=0(119899

119895) 120575

119895

(120590119899minus119895

(119886)) 120585119899minus119895

(55)


119905be a 1-


120585119905

119886 minus 120590119905(119886) 120585

119905

=

infin

sum

119895=1(119905

119895) 120575

119895

(120590119905minus119895) 120585

119905minus119895

(56)


[log 120585 119886] = 119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120590119905(119886) +

infin

sum

119895=1


119895120590minus119895

120575119895

(119886) 120585minus119895

(57)



rarr Ψ119863119874120590

defined by

[log 120585 119886120585119899] = 119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120590119905(119886) 120585

119899

+

infin

sum

119895=1


119895120590minus119895

120575119895

(119886) 120585minus119895

(58)

is a derivation




ΨDO120590= DO

120590oplus INT

120590 (59)

where

DO120590=

119899

sum

119894=0119886119894120585119894

| 119886119894isinA

INT120590=

minus1sum

119894=minusinfin

119886119894120585119894

| 119886119894isinA

(60)







algebras 119863119874120590and 119868119873119879



120590and 119868119873119879

120590

vanish








of ΨDO120590by

ΨDO120590119898

=

119872

sum

119894=119898

119886119894120585119894

| 119886119894isinA (61)




Γ119899= 120572 = (1205721 120572119894 120572119899) 120572119894 isinZ

Γ+

119899= 120572 isin Γ

119899 120572

119894isinZ

+

1le 119894 le 119899

120576119894= (0 0 1 0 0)

120599 =

119899

sum

119894=1120576119894isin Γ

119899

120572 =

119899

prod

119894=1120572119894 120572

119894isin Γ

+

119899

(62)


119899 120575120574(119887) = 12057512057411 ∘ sdot sdot sdot ∘ 120575

120574119899


1205751 120575119899 and

(120572

120574) = prod

119894

(120572119894

120574119894

) (63)


119894 with 119894 = 1 119899




120585119894119886 = 119886120585

119894+ 120575

119894(119886) (64)



119899


119886120585120572

sdot 119887120585120573

= sum

120574isinΓ+

119899

(120572

120574)119886120575

120574

(119887) 120585120572+120573minus120574

(65)


119899DOof formal


120585minus1119894119886 = sum

119895ge0(minus1)119895 120575119895

119894(119886) 120585


(66)


commutator

[119860 119861] = 119860119861minus119861119860

= sum

120574isinΓ+

119899

(1120574)(

120597120574

120597120585120574(119860) 120575

120574

(119861) minus120597120574

120597120585120574(119861) 120575

120574

(119860)) (67)


120597120574

120597120585120574(119886120585

120572

) = 120574 (120572

120574) 119886120585

120572minus120574

120575120574

(119886120585120572

) = 120575120574

(119886) 120585120572

(68)





119894 sdot]

considered in (67)

[log 120585119894 119883] = sum

120574isinΓ+

119899

1120574

120597120574

120597120585120574(log 120585

119894) 120575

120574

(119883)

= sum

120574119894ge0

1120574119894

120597120574

119894

120597120585120574119894

119894

(log 120585119894) 120575

120574119894 (119883)

(69)






[log 120585119894 119860119861] = [log 120585

119894 119860] 119861 +119860 [log 120585

119894 119861] (70)






119894(119886)) =


120591 (119886120575120574

(119887)) = (minus1)120574 120591 (119887120575120574 (119886)) (71)




res Ψ119899119863119874 rarr CΨ

119899119863119874 rarr C defined by


119886120572120585120572

) = 120591 (119886minus120599) (72)

is a trace on Ψ119899119863119874


res (119886120585120572119887120585120573) = res (119887120585120573119886120585120572) (73)


119894 120573

119895 we have 120572

119894 120573

119895ge 0

or 120572119894 120573



119894lt 0

If 120572119894+ 120573


Now suppose that 120572119894+ 120573


119894= 120572

119894+ 120573

119894 We have

res (119886120585120572119887120585120573) = (120572

119896 + 120599) 120591 (119886120575

119896+120599

) (74)


res (119887120585120573119886120585120572) = 120591((120573

119896 + 120599) 119887120575

119896+120599

(119886))

= (119896 minus 120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (minus1)119896+120599 (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

= (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

(75)





119894 119860]119861) 119894 =



119894 sdot] is


119894([119862 119860] 119861) +

119888119894([119861 119862] 119860) = 0




119894= 119898 gt 0 for fixed 1 le 119894 le

119899 and

Ψ119899DO

120572119894

=

sum

120573isinΓ119899

119887120573120585120573

= sum

(1205731 120573119899)isinΓ119899

11988612057312058512057311 sdot sdot sdot 120585

120573119899

119899

isinΨ119899DO 120573

119894ge120572

119894

(76)




120572119894


119899119863119874

120572119894




119899DOlowast

120572119894

≃ sum120579isinΓ119899

119897120579120585120579

120579119894le minus120572

119894minus 1 Let 119860 119861 isin

Ψ119899DO

120572119894



120572119894


120572119894

we have

119888 (119860 119861) = res ([119860 119861] 119871)

= res([

[

sum

120572isinΓ119899

119886120572120585120572

sum

120573isinΓ119899

119887120573120585120573]

]

sum

120579isinΓ119899

120579119894leminus120572119894minus1

119897120579120585120579

)


(])(119887120573) 120575

(120583)

(119897120579)

sdot 120585120572+120573+120579minus(]+120583+120593)


(77)


120572119894+120573

119894+ 120579

119894+ (]

119894+120583

119894+120593

119894) = minus 1 (78)


119888 (119860 119861) = res ([log 120585119894 119860] 119861)

= res([log 120585119894 sum

120572isinΓ119899

119886120572120585120572

] sum

120573isinΓ119899

119887120573120585120573

)


(119886120572) 120575

(120591119894)

(119887120573) 120585

120572+120573minus(120595119894+120591)

)


(79)


120572119894+120573

119894+ (120591

119894+120593

119894) = minus 1 (80)


119894= 120579


120579119894le minus120572

119894minus 1 and 120583

119894ge 0


119894


119899DO



120573isinΓ119899

119887120573120585120573 such that 120573 ge 120572






120572




119894= 120579



Let 119875+119899

= sum120572120582120572119909120572

| 120572 isin Γ+




120582120572119909120572 such that the





119894(119909

120572

) =

120572119894119909120572minus120598119894 with 120572 isin Γ

119899





120572isinΓ+

119899

1120572(119906120575

120572

(V) otimes 120575120572 (119891) 119892) (81)



have the following


119899119863119874 and


Ψ119899119863119874 997888rarr Uotimes119875

119899

119906120585120572

997891997888rarr 119906otimes119909120572

(82)


119899119863119874 andU otimes 119875

119899




+119909120573

minus≃ 119909

120572

otimes119909120573


sum

120572120573isinΓ119899

120582120572120573119909120572

+119909120573

minus (83)


[119909120572

+119909120573

minus 119909

120572

+119909120573

minus] = sum

120574isinΓ+

119899

(1120574) (120597

minusand 120597

+) (119909

120572

+119909120573

minus 119909

120572

+119909120573

minus) (84)


119899as

120597120574

minus(119909

120572

+119909120573

minus) = 120574 (

120573

120574)119909

120572

+119909120573minus120574

minus

120597120574

+(119909

120572

+119909120573

minus) = 120574 (

120572

120574)119909

120572minus120574

+119909120573

minus

(85)


+= 120597

minusotimes 120597

+minus 120597

+otimes 120597

minus


1198910(119909minus120599

) = 1 and 1198910(119909120572

) = 0 if 120572 = minus120599



this section



119899 1198671

(H119899H



minus119894 sdot] 119894 = 1 119899


+= 119909

+1 and 119909minus

= 119909minus1 We








119863 = 120582 [log119909+ sdot] + 120573 [log119909


for some 120582 120573


+) = sum119889

+

119899119898119909119899

+119909119898

minusand

119863(119909minus) = sum119889

minus

119899119898119909119899

+119909119898

minus We have

1 = [119909minus 119909

+] 997904rArr 120582 = [119863 (119909

minus) 119909

+] + [119909

minus 119863 (119909

+)]

997904rArr 120582 = sum119889minus

119899119898[119909

119899

+119909119898

minus 119909

+] +sum119889

+

119899119898[119909

minus 119909

119899

+119909119898

minus]

997904rArr 120582 = sum119889minus

119899119898119898119909

119899


+sum119889+

119899119898119899119909

119899minus1+119909119898

minus

(87)



+) = sum119889

+

0119898119909119898


that119863(119909minus) = sum119889

minus

1198990119909119899


119889minus

119899119898119898119909

119899


= minus119889119899119898119899119909

119899minus1+119909119898

minus(88)


(119898+ 1) 119889minus119899119898+1 = minus (119899 + 1) 119889119899+1119898 (89)


Then

119863(119909+) = sum

119898 =0minus1minus 119889

minus

0119898119909119898


minus1minus+119889

+

00 (90)

119863(119909minus) = sum

119899 =0minus1119889minus

1198990119909119899

++119889

minus10119909minus1++119889

minus

00 (91)

Now we write

119863(119909+) = 120582 [log119909

+ 119909

+] + 120573 [log119909

minus 119909

+] minus [120596 119909

+]


+]

(92)


[120596 119909+] = sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120597minus(120596)

= sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120596

= sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

+119889+

00119909minus +

(93)


119863(119909minus) = 120582 [log119909

+ 119909

minus] + 120573 [log119909

minus 119909

minus] minus [120596 119909

minus]

= 120582119909minus1+minus [120596 119909

minus]

(94)



119899 =0minus1119889minus

1198990119909119899

++minus119889

+

00 997904rArr 120597minus(120596)

= minus sum

119899 =0minus1119889minus

1198990119909119899

+minus119889

+

00 997904rArr 120596

= minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

minus119889minus

00119909+ +120596

(95)


120596 = minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

+ sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

minus119889minus

00119909+ minus119889minus

00119909minus

(96)









119894= res([log 120585

plusmn119894 119860]119861) and







4 Applications



+form a Manin







+and g



minus

vanish



120590 DO

120590 and


form ⟨119860 119861⟩120590



that 119886120572119899





INT119899= 119871 isinΨ

119899DO | ord (119871) lt 0

DO119899= 119871 isinΨ

119899DO | ord (119871) ge 0

(97)



119899oplus DO

119899


119899119863119874 119868119873119879

119899 119863119874








119899DO


INT119894=

(minus1)sum


119886120572120585120572

| 119886120572isin119860

DO119894=

(119899)

sum

(0)isinΓ119899

119886120572120585120572

| 119886120572isin119860

(98)


119894oplusDO

119894


119894 119863119874

119894) formaMa-









119891(A) where Der

119891(A)




(119886 119886) 997891997888rarr (119887 997891997888rarr119891 (120601 (119887) 119886 119886)) (99)



(119887 sdot 120573 ()) = minus120573 (119887) (100)



lowast

(119887 + 119886 +120573) ( + 119886 +120573)

= 119887 + 120601 (119887) 119886 minus 120601 () 119886 + 119886119886 +119908 (119886 119886) + 119887 sdot 120573

minus sdot 120573

(101)


onA119861

119891119861(119887 + 119886 +120573 + 119886 + 120573) = 120573 () + 120573 (119887) +119891 (119886 119886) (102)

Thepair (A119861 119891




119861follows


119908(119886 [1198861015840

119886]) = 119908 (1198861015840

[119886 119886]) minus119908 (119886 [119886 1198861015840

])

119887 sdot 119908 (1198861015840

119886) = 119908 (1198861015840

120601 (119887) sdot 119886) +119908 (120601 (119887) sdot 1198861015840

119886)

(103)





lowast with bracket

[(1198871 1198711 1205731) (1198872 1198712 1205732)] = (0 1198871 [log 120585 1198712]

minus 1198872 [log 120585 1198711] + [1198711 1198712] ⟨[log 120585 1198711] 1198712⟩)(104)


⟨(1198871 1198711 1205731) (1198872 1198712 1205732)⟩

= 12057311198872 +12057321198871 + res (11987111198712) (105)




120590)



120590oplus INT

120590) in which

DO = 120572119888 +

119899

sum

119894=0119886119894120585119894

| 119886119894isinA 120572 isinR


119894=minusinfin

119886119894120585119894

| 119886119894isinA 120573 isinR

(106)



(Ψ119863119874119863119874 119868119873119879)

(Ψ120590119863119874119863119874

120590 119868119873119879

120590)

(107)

are Manin triples




119899DO = Ψ


119894



Ψ119899DO = INT

119894oplus DO

119894 (108)

where

INT119894=

120573 log 120585119894+

(minus1)sum


119886120579120585120579

| 119886120579isin119860

DO119894=

120572119888119894+

(119899)

sum

(0)isinΓ119899

119886120579120585120579

| 119886120579isin119860

(109)



119894 119863119874

119894) form a Ma-






decomposition

Ψ119899DO = Ψ




[(119860 120572 120573) (119861 120572 120573)] = ([119860 119861] + [120573 sdot log 120585 119860]

minus [120573 sdot log 120585 119861] 119888 (119860 119861) 0) (111)

and bilinear form

⟨119860 119861⟩ = res (119860 119861) + 120572 sdot 120573 +120573 sdot 120572 (112)


119899) 0 = (0 0) isin R119899 and 119888(119860 119861) = (1198881(119860 119861)





⟨]120575119865

120575120583⟩ =

119889

119889120598

10038161003816100381610038161003816100381610038161003816120598=0119865 (120583 + 120598]) (113)



119865 119866 (120583) = ⟨120583 [120575119865

120575120583120575119866

120575120583]⟩ (114)



119889120583

119889119905= ⟨120583 [sdot

120575119867

120575120583]⟩ (115)


119877on g by

[119875 119876]119877= [119877 (119875) 119876] + [119875 119877 (119876)] 119876 119875 isin g (116)





+minus



plusmn We write

119875plusmn= 120587


119877= [119875

+ 119876

+]minus[119875

minus 119876

minus] and

119865 119866119877= ⟨120583 [

120575119865

120575120583120575119866

120575120583]

119877

⟩ (117)


⟨120583 [120575119865

120575120583 sdot]⟩ = 0 (118)


119865 119866119877= ⟨120583 [(

120575119865

120575120583)

+

120575119866

120575120583]⟩ (119)


119883119867(120583) sdot 119865 = ⟨120583 [minus(

120575119867

120575120583)

+

120575119865

120575120583]⟩ (120)


⟨119875 [119876 119877]⟩ = ⟨[119877 119875] 119876⟩ (121)


119889119875

119889119905= [(

120575119867

120575120583)

+

119875] (122)



⟨[119875 119876] 119878⟩ = ⟨[119878 119875] 119876⟩ 119878 119875 119876 isin ΨDO (123)


minusoplus ΨDO

+allows us to




Ψ119863119874 Then

120575119867119896

120575119871= 119896119871

119896minus1 (124)


119867119896 119867

119897119877(119871) = 0 for any 119877-matrix 119877




119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (125)


120590 DO

120590 and INT

120590


120590as



120590 The


119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (126)



119899DO DO

119899 and INT

119899allows us


[ ]119877 Once more we set 119867

119896(119871) = res(119871119896) for 119871 isin Ψ

119899DO


120597119871

120597119905= [(119896119871

119896minus1)+

119871] (127)




⟨(119860 120572) (119861 120573)⟩ = ⟨119860 119861⟩ + 120572120573 = res (119860119861) + 120572120573 (128)




⟨120597119871

120597119905 119876⟩ = ⟨[

120597119867

120597119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (129)

119889119897

119889119905= 0 (130)



119889 ( )

119889119905sdot (119876 119903) = ⟨( ) [(119876 119903)

120575119867

120575 ( )]⟩

= ⟨( ) [(119876 119903) (120575119867

120575120575119867

120575)]⟩

= ⟨( ) ([119876120575119867

120575] 119888 (119876

120575119867

120575))⟩

(131)


⟨120597 (119871 119897)

120597119905 (119876 119903)⟩

= ⟨(119871 119897) ([119876120575119867

120575119871] 119888 (119876

120575119867

120575119871))⟩

= ⟨119871 [119876120575119867

120575119871]⟩+ 119897119888 (119876

120575119867

120575119871)

(132)


⟨120597119871

120597119905 119876⟩+ 119903

120597119897

120597119905= ⟨[

120575119867

120575119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (133)





res ([log 120585 119886120585] 119887120585) = res (119887120585 [log 120585 119886120585])

= res(119887120585infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 1205851minus119895)

= res(infin

sum

119895=0

(minus1)119895+1

119895(119887120575

119895

(119886) 1205852minus119895

+ 119887120575119895+1

(119886) 1205851minus119895))

=(minus1)3+1

3119887120575

3(119886) +

(minus1)2+1

2119887120575

3(119886)

= minus16120591 (119887120575

3(119886))

(134)


⟨(119886120585 120572) (119887120585 120573)⟩ = 120591 (119886119887) + 120572120573 (135)


⟨119889119906

119889119905120585 119887120585⟩ = ⟨[ℎ (119906) 120585 119906120585] 119887120585⟩

minus16119897120591 (119887120575

3(ℎ (119906)))

(136)


119889119906

119889119905= ℎ (119906) 120575 (119906) minus 119906120575 (ℎ (119906)) minus (

16) 119897120575

3(ℎ (119906)) (137)




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119902 119862

119902


the formula

119889119888 (1198921 119892119902+1) = sum


sdot 119888 ([119892119904 119892

119905] 1198921 119892119904 119892119905 119892119902+1)

+ sum

1le119904le119902+1(minus1)119904 119892

119904119888 (1198921 119892119904 119892119902+1)

(22)







(g)






[(119883 119906) (119884 V)] = ([119883 119884] 119888 (119883 119884)) (23)




119888 ([119883 119884] 119885) + 119888 ([119885119883] 119884) + 119888 ([119884 119885] 119883) = 0 (24)




(gn)







120595 (119909) sdot 119909 = 0 (25)


1(g)







120595+[119886 sdot]


1198671(g) if and only if ⟨120575

120595(119909) 119909⟩ = 0 for all 119909 isin g



119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120585119905

= log 120585 (26)


119905=0 (119905

119895) = (minus1)119895+1119895 we obtain

[log 120585 119886120585119899] =infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 120585119899minus119895

(27)





119904(120573 minus 119904

119896) =

119904+119896

sum

119895=119904


119895(

120573

119904 + 119896 minus 119895)(

119895

119895 minus 119904) (28)








[log 120585 119886120585119899 ∘ 119887120585119898] = [log 120585 119886120585119899] ∘ 119887120585119898 + 119886120585119899

∘ [log 120585 119887120585119898] (29)


[log 120585 119886120585119899 ∘ 119887120585119898] = [log 120585infin

sum

119896=119900

(119899

119896)119886120575

119895

(119887) 120585119899+119898minus119896

]

=

infin

sum

119896=119900

infin

sum

119895=1(119899

119896)(minus1)119895+1

119895120575119895

(119886120575119896

(119887)) 120585119899+119898minus119896minus119895

(30)


[log 120585 119886120585119899] ∘ 119887120585119898 + 119886120585119899 ∘ [log 120585 119887120585119898]

= (

infin

sum

119895=119900

(minus1)119895+1

119895120575119895

(119886) 120585119899minus119895

)119887120585119898

+ 119886120585119899

(

infin

sum

119895=119900

(minus1)119895+1

119895120575119895

(119887) 120585119898minus119895

)

=

infin

sum

119895=1

infin

sum

119896=0(119899 minus 119895

119896)(minus1)119895+1

119895120575119895

(119886) 120575119896

(119887) 120585119899+119898minus119896minus119895

+

infin

sum

119895=1

infin

sum

119896=0(119899

119896)(minus1)119895+1

119895119886120575

119896

(120575119895

(119887)) 120585119899+119898minus119896minus119895

=

infin

sum

119895=1

infin

sum

119896=0(119899 minus 119895

119896)(minus1)119895+1

119895120575119895

(119886) 120575119896

(119887) 120585119899+119898minus119896minus119895

+

infin

sum

119895=1

infin

sum

119896=0(119899

119896)(minus1)119895+1

119895119886120575

119896+119895

(119887) 120585119899+119898minus119896minus119895

(31)


sum(119899

119896)(minus1)119895+1

119895120575119895

(119886120575119896

(119887)) (32)


sum

119895

sum

119894=119900

(119899

119896)(

119894

119895)(minus1)119895+1

119895120575119895minus119894

(119886) 120575119894+119896

(119887)

= sum

119895minus1

sum

119894=119900

(119899

119896)(

119894

119895)(minus1)119895+1

119895120575119895minus119894

(119886) 120575119894+119896

(119887)

+sum(119899

119896)(minus1)119895+1

119895119886120575

119903

(119887)

(33)


sum(119899 minus 119904

119897)(minus1)119904+1

119904120575119904

(119886) 120575119897

(119887)

+sum(119899

119897)(minus1)119904+1

119904119886120575

119903

(119887)

(34)


sum(119899

119896)(

119895

119894)(minus1)119895+1

119895= (

119899 minus 119904

119897)(minus1)119904+1

119904 (35)


119897+119904

sum

119895=119904

(119899

119897 + 119904 minus 119895)(

119895

119895 minus 119904)(minus1)119895+1

119895= (

119899 minus 119904

119897)(minus1)119904+1

119904 (36)



119888 (119860 119861) = res ([log 120585 119860] ∘ 119861) (37)



res ([log 120585 119886120585119899]) = 120591 ((minus1)119899

119899 + 1120575119899+1

(119886)) = 0 (38)


119888 ([119861 119860]) = res ([log 120585 119861]119860)

= res ([log 120585 119861119860] minus119861 [log 120585 119860])

= minus 119888 (119860 119861)

(39)




119888 (119891 (119909) 120597 119892 (119909) 120597) =16int119891

1015840

(119909) 11989210158401015840

(119909) 119889119909 (40)



119888 (119891120597119899

119892120597119898

) =119899119898

(119898 + 119899 + 1)int1198781119891(119898+1)

119892(119899)

119889120579 (41)


119888 (119860 119861) = int1198781([119909 119860] ∘ 119861) (42)



1015840


119872ΨDO =

119899

sum

119894=minusinfin

119886119894119863V

119894

| 119886119894isinM (43)


119863(sum

119899

119894=minusinfin119886119894119863119894

V) = 119886minus1119863



119875(res

119863(119860))Then as










119898≃ sum


119886119894120585119894



119894=119898119886119894120585119894 119861 =

sum119872

119894=119898119887119894120585119894



119897119894120585119894

isin ΨDOlowast

119898we

have

119888 (119860 119861) = res ([119860 119861] 119871) = res([

[

119872

sum

119894=119898

119886119894120585119894

119872

sum

119895=119898

119887119895120585119895]

]

sdot


119896=minusinfin

119897119896120585119896

)

= res( sum


119894)

sdot 120575(119905)

(119887119895) 120575

(119906)

(119897119896) 120585

119894+119895+119896minus(119905+119906+119904)

))


(44)


119894 + 119895 + 119896 minus (119905 + 119906 + 119904) = minus 1 (45)


119888 (119860 119861) = res ([log 120585 119860] 119861) = res([log 120585119872

sum

119894=119898

119886119894120585119894

]

sdot

119872

sum

119895=119898

119887119895120585119895

) = res( sum

119894119895119905119904=11989811989800(coefficient) 120575(119904) (119886

119894)

sdot 120575(119905)

(119887119895) 120585

119894+119895minus(119904+119905)


(46)


119894 + 119895 minus (119904 + 119905) = minus 1 (47)



119898









ΨDO120590=

119873

sum

119894=minusinfin

119886119894120585119894

119873 isinZ 119886119899isinA (48)


120585119886 = 120590 (119886) 120585 + 120575 (119886)

120585120585minus1= 120585

minus1120585 = 1

(49)


120585119899

119886 =

119899

sum

119894=0119875119894119899(120590 120575) (119886) 120585

119894

(50)





120585119899

119886 =

infin

sum


infin

sum


sdot 120590minus1(120575120590

minus1)119894119899


minus1)1198941(119886) 120585


(51)



119898=minusinfin119887119898120585119898

then

119860119861 =

119872

sum

119898=minusinfin

sum

119899lt0sum

119894ge0(minus1)|119894| 119886

119899120590minus1(120575120590



minus1)1198941(119887119898) 120585

119898+119899+|119894|

+

119872

sum

119898=minusinfin

119873

sum

119899=0

119899

sum

119895=0119886119899119875119895119899(120590 120575) (119887

119898) 120585

119898+119895

(52)




120591 (119887120590minus1(120575120590


minus1(120575120590

minus1)119894119898

(119886))




res(119899

sum

119894=minusinfin

119886119894120585119894

) = 120591 (119886minus1) (54)

is a trace on Ψ119863119874120590


120585119899

119886 =

infin

sum

119895=0(119899

119895) 120575

119895

(120590119899minus119895

(119886)) 120585119899minus119895

(55)


119905be a 1-


120585119905

119886 minus 120590119905(119886) 120585

119905

=

infin

sum

119895=1(119905

119895) 120575

119895

(120590119905minus119895) 120585

119905minus119895

(56)


[log 120585 119886] = 119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120590119905(119886) +

infin

sum

119895=1


119895120590minus119895

120575119895

(119886) 120585minus119895

(57)



rarr Ψ119863119874120590

defined by

[log 120585 119886120585119899] = 119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120590119905(119886) 120585

119899

+

infin

sum

119895=1


119895120590minus119895

120575119895

(119886) 120585minus119895

(58)

is a derivation




ΨDO120590= DO

120590oplus INT

120590 (59)

where

DO120590=

119899

sum

119894=0119886119894120585119894

| 119886119894isinA

INT120590=

minus1sum

119894=minusinfin

119886119894120585119894

| 119886119894isinA

(60)







algebras 119863119874120590and 119868119873119879



120590and 119868119873119879

120590

vanish








of ΨDO120590by

ΨDO120590119898

=

119872

sum

119894=119898

119886119894120585119894

| 119886119894isinA (61)




Γ119899= 120572 = (1205721 120572119894 120572119899) 120572119894 isinZ

Γ+

119899= 120572 isin Γ

119899 120572

119894isinZ

+

1le 119894 le 119899

120576119894= (0 0 1 0 0)

120599 =

119899

sum

119894=1120576119894isin Γ

119899

120572 =

119899

prod

119894=1120572119894 120572

119894isin Γ

+

119899

(62)


119899 120575120574(119887) = 12057512057411 ∘ sdot sdot sdot ∘ 120575

120574119899


1205751 120575119899 and

(120572

120574) = prod

119894

(120572119894

120574119894

) (63)


119894 with 119894 = 1 119899




120585119894119886 = 119886120585

119894+ 120575

119894(119886) (64)



119899


119886120585120572

sdot 119887120585120573

= sum

120574isinΓ+

119899

(120572

120574)119886120575

120574

(119887) 120585120572+120573minus120574

(65)


119899DOof formal


120585minus1119894119886 = sum

119895ge0(minus1)119895 120575119895

119894(119886) 120585


(66)


commutator

[119860 119861] = 119860119861minus119861119860

= sum

120574isinΓ+

119899

(1120574)(

120597120574

120597120585120574(119860) 120575

120574

(119861) minus120597120574

120597120585120574(119861) 120575

120574

(119860)) (67)


120597120574

120597120585120574(119886120585

120572

) = 120574 (120572

120574) 119886120585

120572minus120574

120575120574

(119886120585120572

) = 120575120574

(119886) 120585120572

(68)





119894 sdot]

considered in (67)

[log 120585119894 119883] = sum

120574isinΓ+

119899

1120574

120597120574

120597120585120574(log 120585

119894) 120575

120574

(119883)

= sum

120574119894ge0

1120574119894

120597120574

119894

120597120585120574119894

119894

(log 120585119894) 120575

120574119894 (119883)

(69)






[log 120585119894 119860119861] = [log 120585

119894 119860] 119861 +119860 [log 120585

119894 119861] (70)






119894(119886)) =


120591 (119886120575120574

(119887)) = (minus1)120574 120591 (119887120575120574 (119886)) (71)




res Ψ119899119863119874 rarr CΨ

119899119863119874 rarr C defined by


119886120572120585120572

) = 120591 (119886minus120599) (72)

is a trace on Ψ119899119863119874


res (119886120585120572119887120585120573) = res (119887120585120573119886120585120572) (73)


119894 120573

119895 we have 120572

119894 120573

119895ge 0

or 120572119894 120573



119894lt 0

If 120572119894+ 120573


Now suppose that 120572119894+ 120573


119894= 120572

119894+ 120573

119894 We have

res (119886120585120572119887120585120573) = (120572

119896 + 120599) 120591 (119886120575

119896+120599

) (74)


res (119887120585120573119886120585120572) = 120591((120573

119896 + 120599) 119887120575

119896+120599

(119886))

= (119896 minus 120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (minus1)119896+120599 (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

= (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

(75)





119894 119860]119861) 119894 =



119894 sdot] is


119894([119862 119860] 119861) +

119888119894([119861 119862] 119860) = 0




119894= 119898 gt 0 for fixed 1 le 119894 le

119899 and

Ψ119899DO

120572119894

=

sum

120573isinΓ119899

119887120573120585120573

= sum

(1205731 120573119899)isinΓ119899

11988612057312058512057311 sdot sdot sdot 120585

120573119899

119899

isinΨ119899DO 120573

119894ge120572

119894

(76)




120572119894


119899119863119874

120572119894




119899DOlowast

120572119894

≃ sum120579isinΓ119899

119897120579120585120579

120579119894le minus120572

119894minus 1 Let 119860 119861 isin

Ψ119899DO

120572119894



120572119894


120572119894

we have

119888 (119860 119861) = res ([119860 119861] 119871)

= res([

[

sum

120572isinΓ119899

119886120572120585120572

sum

120573isinΓ119899

119887120573120585120573]

]

sum

120579isinΓ119899

120579119894leminus120572119894minus1

119897120579120585120579

)


(])(119887120573) 120575

(120583)

(119897120579)

sdot 120585120572+120573+120579minus(]+120583+120593)


(77)


120572119894+120573

119894+ 120579

119894+ (]

119894+120583

119894+120593

119894) = minus 1 (78)


119888 (119860 119861) = res ([log 120585119894 119860] 119861)

= res([log 120585119894 sum

120572isinΓ119899

119886120572120585120572

] sum

120573isinΓ119899

119887120573120585120573

)


(119886120572) 120575

(120591119894)

(119887120573) 120585

120572+120573minus(120595119894+120591)

)


(79)


120572119894+120573

119894+ (120591

119894+120593

119894) = minus 1 (80)


119894= 120579


120579119894le minus120572

119894minus 1 and 120583

119894ge 0


119894


119899DO



120573isinΓ119899

119887120573120585120573 such that 120573 ge 120572






120572




119894= 120579



Let 119875+119899

= sum120572120582120572119909120572

| 120572 isin Γ+




120582120572119909120572 such that the





119894(119909

120572

) =

120572119894119909120572minus120598119894 with 120572 isin Γ

119899





120572isinΓ+

119899

1120572(119906120575

120572

(V) otimes 120575120572 (119891) 119892) (81)



have the following


119899119863119874 and


Ψ119899119863119874 997888rarr Uotimes119875

119899

119906120585120572

997891997888rarr 119906otimes119909120572

(82)


119899119863119874 andU otimes 119875

119899




+119909120573

minus≃ 119909

120572

otimes119909120573


sum

120572120573isinΓ119899

120582120572120573119909120572

+119909120573

minus (83)


[119909120572

+119909120573

minus 119909

120572

+119909120573

minus] = sum

120574isinΓ+

119899

(1120574) (120597

minusand 120597

+) (119909

120572

+119909120573

minus 119909

120572

+119909120573

minus) (84)


119899as

120597120574

minus(119909

120572

+119909120573

minus) = 120574 (

120573

120574)119909

120572

+119909120573minus120574

minus

120597120574

+(119909

120572

+119909120573

minus) = 120574 (

120572

120574)119909

120572minus120574

+119909120573

minus

(85)


+= 120597

minusotimes 120597

+minus 120597

+otimes 120597

minus


1198910(119909minus120599

) = 1 and 1198910(119909120572

) = 0 if 120572 = minus120599



this section



119899 1198671

(H119899H



minus119894 sdot] 119894 = 1 119899


+= 119909

+1 and 119909minus

= 119909minus1 We








119863 = 120582 [log119909+ sdot] + 120573 [log119909


for some 120582 120573


+) = sum119889

+

119899119898119909119899

+119909119898

minusand

119863(119909minus) = sum119889

minus

119899119898119909119899

+119909119898

minus We have

1 = [119909minus 119909

+] 997904rArr 120582 = [119863 (119909

minus) 119909

+] + [119909

minus 119863 (119909

+)]

997904rArr 120582 = sum119889minus

119899119898[119909

119899

+119909119898

minus 119909

+] +sum119889

+

119899119898[119909

minus 119909

119899

+119909119898

minus]

997904rArr 120582 = sum119889minus

119899119898119898119909

119899


+sum119889+

119899119898119899119909

119899minus1+119909119898

minus

(87)



+) = sum119889

+

0119898119909119898


that119863(119909minus) = sum119889

minus

1198990119909119899


119889minus

119899119898119898119909

119899


= minus119889119899119898119899119909

119899minus1+119909119898

minus(88)


(119898+ 1) 119889minus119899119898+1 = minus (119899 + 1) 119889119899+1119898 (89)


Then

119863(119909+) = sum

119898 =0minus1minus 119889

minus

0119898119909119898


minus1minus+119889

+

00 (90)

119863(119909minus) = sum

119899 =0minus1119889minus

1198990119909119899

++119889

minus10119909minus1++119889

minus

00 (91)

Now we write

119863(119909+) = 120582 [log119909

+ 119909

+] + 120573 [log119909

minus 119909

+] minus [120596 119909

+]


+]

(92)


[120596 119909+] = sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120597minus(120596)

= sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120596

= sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

+119889+

00119909minus +

(93)


119863(119909minus) = 120582 [log119909

+ 119909

minus] + 120573 [log119909

minus 119909

minus] minus [120596 119909

minus]

= 120582119909minus1+minus [120596 119909

minus]

(94)



119899 =0minus1119889minus

1198990119909119899

++minus119889

+

00 997904rArr 120597minus(120596)

= minus sum

119899 =0minus1119889minus

1198990119909119899

+minus119889

+

00 997904rArr 120596

= minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

minus119889minus

00119909+ +120596

(95)


120596 = minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

+ sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

minus119889minus

00119909+ minus119889minus

00119909minus

(96)









119894= res([log 120585

plusmn119894 119860]119861) and







4 Applications



+form a Manin







+and g



minus

vanish



120590 DO

120590 and


form ⟨119860 119861⟩120590



that 119886120572119899





INT119899= 119871 isinΨ

119899DO | ord (119871) lt 0

DO119899= 119871 isinΨ

119899DO | ord (119871) ge 0

(97)



119899oplus DO

119899


119899119863119874 119868119873119879

119899 119863119874








119899DO


INT119894=

(minus1)sum


119886120572120585120572

| 119886120572isin119860

DO119894=

(119899)

sum

(0)isinΓ119899

119886120572120585120572

| 119886120572isin119860

(98)


119894oplusDO

119894


119894 119863119874

119894) formaMa-









119891(A) where Der

119891(A)




(119886 119886) 997891997888rarr (119887 997891997888rarr119891 (120601 (119887) 119886 119886)) (99)



(119887 sdot 120573 ()) = minus120573 (119887) (100)



lowast

(119887 + 119886 +120573) ( + 119886 +120573)

= 119887 + 120601 (119887) 119886 minus 120601 () 119886 + 119886119886 +119908 (119886 119886) + 119887 sdot 120573

minus sdot 120573

(101)


onA119861

119891119861(119887 + 119886 +120573 + 119886 + 120573) = 120573 () + 120573 (119887) +119891 (119886 119886) (102)

Thepair (A119861 119891




119861follows


119908(119886 [1198861015840

119886]) = 119908 (1198861015840

[119886 119886]) minus119908 (119886 [119886 1198861015840

])

119887 sdot 119908 (1198861015840

119886) = 119908 (1198861015840

120601 (119887) sdot 119886) +119908 (120601 (119887) sdot 1198861015840

119886)

(103)





lowast with bracket

[(1198871 1198711 1205731) (1198872 1198712 1205732)] = (0 1198871 [log 120585 1198712]

minus 1198872 [log 120585 1198711] + [1198711 1198712] ⟨[log 120585 1198711] 1198712⟩)(104)


⟨(1198871 1198711 1205731) (1198872 1198712 1205732)⟩

= 12057311198872 +12057321198871 + res (11987111198712) (105)




120590)



120590oplus INT

120590) in which

DO = 120572119888 +

119899

sum

119894=0119886119894120585119894

| 119886119894isinA 120572 isinR


119894=minusinfin

119886119894120585119894

| 119886119894isinA 120573 isinR

(106)



(Ψ119863119874119863119874 119868119873119879)

(Ψ120590119863119874119863119874

120590 119868119873119879

120590)

(107)

are Manin triples




119899DO = Ψ


119894



Ψ119899DO = INT

119894oplus DO

119894 (108)

where

INT119894=

120573 log 120585119894+

(minus1)sum


119886120579120585120579

| 119886120579isin119860

DO119894=

120572119888119894+

(119899)

sum

(0)isinΓ119899

119886120579120585120579

| 119886120579isin119860

(109)



119894 119863119874

119894) form a Ma-






decomposition

Ψ119899DO = Ψ




[(119860 120572 120573) (119861 120572 120573)] = ([119860 119861] + [120573 sdot log 120585 119860]

minus [120573 sdot log 120585 119861] 119888 (119860 119861) 0) (111)

and bilinear form

⟨119860 119861⟩ = res (119860 119861) + 120572 sdot 120573 +120573 sdot 120572 (112)


119899) 0 = (0 0) isin R119899 and 119888(119860 119861) = (1198881(119860 119861)





⟨]120575119865

120575120583⟩ =

119889

119889120598

10038161003816100381610038161003816100381610038161003816120598=0119865 (120583 + 120598]) (113)



119865 119866 (120583) = ⟨120583 [120575119865

120575120583120575119866

120575120583]⟩ (114)



119889120583

119889119905= ⟨120583 [sdot

120575119867

120575120583]⟩ (115)


119877on g by

[119875 119876]119877= [119877 (119875) 119876] + [119875 119877 (119876)] 119876 119875 isin g (116)





+minus



plusmn We write

119875plusmn= 120587


119877= [119875

+ 119876

+]minus[119875

minus 119876

minus] and

119865 119866119877= ⟨120583 [

120575119865

120575120583120575119866

120575120583]

119877

⟩ (117)


⟨120583 [120575119865

120575120583 sdot]⟩ = 0 (118)


119865 119866119877= ⟨120583 [(

120575119865

120575120583)

+

120575119866

120575120583]⟩ (119)


119883119867(120583) sdot 119865 = ⟨120583 [minus(

120575119867

120575120583)

+

120575119865

120575120583]⟩ (120)


⟨119875 [119876 119877]⟩ = ⟨[119877 119875] 119876⟩ (121)


119889119875

119889119905= [(

120575119867

120575120583)

+

119875] (122)



⟨[119875 119876] 119878⟩ = ⟨[119878 119875] 119876⟩ 119878 119875 119876 isin ΨDO (123)


minusoplus ΨDO

+allows us to




Ψ119863119874 Then

120575119867119896

120575119871= 119896119871

119896minus1 (124)


119867119896 119867

119897119877(119871) = 0 for any 119877-matrix 119877




119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (125)


120590 DO

120590 and INT

120590


120590as



120590 The


119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (126)



119899DO DO

119899 and INT

119899allows us


[ ]119877 Once more we set 119867

119896(119871) = res(119871119896) for 119871 isin Ψ

119899DO


120597119871

120597119905= [(119896119871

119896minus1)+

119871] (127)




⟨(119860 120572) (119861 120573)⟩ = ⟨119860 119861⟩ + 120572120573 = res (119860119861) + 120572120573 (128)




⟨120597119871

120597119905 119876⟩ = ⟨[

120597119867

120597119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (129)

119889119897

119889119905= 0 (130)



119889 ( )

119889119905sdot (119876 119903) = ⟨( ) [(119876 119903)

120575119867

120575 ( )]⟩

= ⟨( ) [(119876 119903) (120575119867

120575120575119867

120575)]⟩

= ⟨( ) ([119876120575119867

120575] 119888 (119876

120575119867

120575))⟩

(131)


⟨120597 (119871 119897)

120597119905 (119876 119903)⟩

= ⟨(119871 119897) ([119876120575119867

120575119871] 119888 (119876

120575119867

120575119871))⟩

= ⟨119871 [119876120575119867

120575119871]⟩+ 119897119888 (119876

120575119867

120575119871)

(132)


⟨120597119871

120597119905 119876⟩+ 119903

120597119897

120597119905= ⟨[

120575119867

120575119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (133)





res ([log 120585 119886120585] 119887120585) = res (119887120585 [log 120585 119886120585])

= res(119887120585infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 1205851minus119895)

= res(infin

sum

119895=0

(minus1)119895+1

119895(119887120575

119895

(119886) 1205852minus119895

+ 119887120575119895+1

(119886) 1205851minus119895))

=(minus1)3+1

3119887120575

3(119886) +

(minus1)2+1

2119887120575

3(119886)

= minus16120591 (119887120575

3(119886))

(134)


⟨(119886120585 120572) (119887120585 120573)⟩ = 120591 (119886119887) + 120572120573 (135)


⟨119889119906

119889119905120585 119887120585⟩ = ⟨[ℎ (119906) 120585 119906120585] 119887120585⟩

minus16119897120591 (119887120575

3(ℎ (119906)))

(136)


119889119906

119889119905= ℎ (119906) 120575 (119906) minus 119906120575 (ℎ (119906)) minus (

16) 119897120575

3(ℎ (119906)) (137)




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















[log 120585 119886120585119899 ∘ 119887120585119898] = [log 120585 119886120585119899] ∘ 119887120585119898 + 119886120585119899

∘ [log 120585 119887120585119898] (29)


[log 120585 119886120585119899 ∘ 119887120585119898] = [log 120585infin

sum

119896=119900

(119899

119896)119886120575

119895

(119887) 120585119899+119898minus119896

]

=

infin

sum

119896=119900

infin

sum

119895=1(119899

119896)(minus1)119895+1

119895120575119895

(119886120575119896

(119887)) 120585119899+119898minus119896minus119895

(30)


[log 120585 119886120585119899] ∘ 119887120585119898 + 119886120585119899 ∘ [log 120585 119887120585119898]

= (

infin

sum

119895=119900

(minus1)119895+1

119895120575119895

(119886) 120585119899minus119895

)119887120585119898

+ 119886120585119899

(

infin

sum

119895=119900

(minus1)119895+1

119895120575119895

(119887) 120585119898minus119895

)

=

infin

sum

119895=1

infin

sum

119896=0(119899 minus 119895

119896)(minus1)119895+1

119895120575119895

(119886) 120575119896

(119887) 120585119899+119898minus119896minus119895

+

infin

sum

119895=1

infin

sum

119896=0(119899

119896)(minus1)119895+1

119895119886120575

119896

(120575119895

(119887)) 120585119899+119898minus119896minus119895

=

infin

sum

119895=1

infin

sum

119896=0(119899 minus 119895

119896)(minus1)119895+1

119895120575119895

(119886) 120575119896

(119887) 120585119899+119898minus119896minus119895

+

infin

sum

119895=1

infin

sum

119896=0(119899

119896)(minus1)119895+1

119895119886120575

119896+119895

(119887) 120585119899+119898minus119896minus119895

(31)


sum(119899

119896)(minus1)119895+1

119895120575119895

(119886120575119896

(119887)) (32)


sum

119895

sum

119894=119900

(119899

119896)(

119894

119895)(minus1)119895+1

119895120575119895minus119894

(119886) 120575119894+119896

(119887)

= sum

119895minus1

sum

119894=119900

(119899

119896)(

119894

119895)(minus1)119895+1

119895120575119895minus119894

(119886) 120575119894+119896

(119887)

+sum(119899

119896)(minus1)119895+1

119895119886120575

119903

(119887)

(33)


sum(119899 minus 119904

119897)(minus1)119904+1

119904120575119904

(119886) 120575119897

(119887)

+sum(119899

119897)(minus1)119904+1

119904119886120575

119903

(119887)

(34)


sum(119899

119896)(

119895

119894)(minus1)119895+1

119895= (

119899 minus 119904

119897)(minus1)119904+1

119904 (35)


119897+119904

sum

119895=119904

(119899

119897 + 119904 minus 119895)(

119895

119895 minus 119904)(minus1)119895+1

119895= (

119899 minus 119904

119897)(minus1)119904+1

119904 (36)



119888 (119860 119861) = res ([log 120585 119860] ∘ 119861) (37)



res ([log 120585 119886120585119899]) = 120591 ((minus1)119899

119899 + 1120575119899+1

(119886)) = 0 (38)


119888 ([119861 119860]) = res ([log 120585 119861]119860)

= res ([log 120585 119861119860] minus119861 [log 120585 119860])

= minus 119888 (119860 119861)

(39)




119888 (119891 (119909) 120597 119892 (119909) 120597) =16int119891

1015840

(119909) 11989210158401015840

(119909) 119889119909 (40)



119888 (119891120597119899

119892120597119898

) =119899119898

(119898 + 119899 + 1)int1198781119891(119898+1)

119892(119899)

119889120579 (41)


119888 (119860 119861) = int1198781([119909 119860] ∘ 119861) (42)



1015840


119872ΨDO =

119899

sum

119894=minusinfin

119886119894119863V

119894

| 119886119894isinM (43)


119863(sum

119899

119894=minusinfin119886119894119863119894

V) = 119886minus1119863



119875(res

119863(119860))Then as










119898≃ sum


119886119894120585119894



119894=119898119886119894120585119894 119861 =

sum119872

119894=119898119887119894120585119894



119897119894120585119894

isin ΨDOlowast

119898we

have

119888 (119860 119861) = res ([119860 119861] 119871) = res([

[

119872

sum

119894=119898

119886119894120585119894

119872

sum

119895=119898

119887119895120585119895]

]

sdot


119896=minusinfin

119897119896120585119896

)

= res( sum


119894)

sdot 120575(119905)

(119887119895) 120575

(119906)

(119897119896) 120585

119894+119895+119896minus(119905+119906+119904)

))


(44)


119894 + 119895 + 119896 minus (119905 + 119906 + 119904) = minus 1 (45)


119888 (119860 119861) = res ([log 120585 119860] 119861) = res([log 120585119872

sum

119894=119898

119886119894120585119894

]

sdot

119872

sum

119895=119898

119887119895120585119895

) = res( sum

119894119895119905119904=11989811989800(coefficient) 120575(119904) (119886

119894)

sdot 120575(119905)

(119887119895) 120585

119894+119895minus(119904+119905)


(46)


119894 + 119895 minus (119904 + 119905) = minus 1 (47)



119898









ΨDO120590=

119873

sum

119894=minusinfin

119886119894120585119894

119873 isinZ 119886119899isinA (48)


120585119886 = 120590 (119886) 120585 + 120575 (119886)

120585120585minus1= 120585

minus1120585 = 1

(49)


120585119899

119886 =

119899

sum

119894=0119875119894119899(120590 120575) (119886) 120585

119894

(50)





120585119899

119886 =

infin

sum


infin

sum


sdot 120590minus1(120575120590

minus1)119894119899


minus1)1198941(119886) 120585


(51)



119898=minusinfin119887119898120585119898

then

119860119861 =

119872

sum

119898=minusinfin

sum

119899lt0sum

119894ge0(minus1)|119894| 119886

119899120590minus1(120575120590



minus1)1198941(119887119898) 120585

119898+119899+|119894|

+

119872

sum

119898=minusinfin

119873

sum

119899=0

119899

sum

119895=0119886119899119875119895119899(120590 120575) (119887

119898) 120585

119898+119895

(52)




120591 (119887120590minus1(120575120590


minus1(120575120590

minus1)119894119898

(119886))




res(119899

sum

119894=minusinfin

119886119894120585119894

) = 120591 (119886minus1) (54)

is a trace on Ψ119863119874120590


120585119899

119886 =

infin

sum

119895=0(119899

119895) 120575

119895

(120590119899minus119895

(119886)) 120585119899minus119895

(55)


119905be a 1-


120585119905

119886 minus 120590119905(119886) 120585

119905

=

infin

sum

119895=1(119905

119895) 120575

119895

(120590119905minus119895) 120585

119905minus119895

(56)


[log 120585 119886] = 119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120590119905(119886) +

infin

sum

119895=1


119895120590minus119895

120575119895

(119886) 120585minus119895

(57)



rarr Ψ119863119874120590

defined by

[log 120585 119886120585119899] = 119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120590119905(119886) 120585

119899

+

infin

sum

119895=1


119895120590minus119895

120575119895

(119886) 120585minus119895

(58)

is a derivation




ΨDO120590= DO

120590oplus INT

120590 (59)

where

DO120590=

119899

sum

119894=0119886119894120585119894

| 119886119894isinA

INT120590=

minus1sum

119894=minusinfin

119886119894120585119894

| 119886119894isinA

(60)







algebras 119863119874120590and 119868119873119879



120590and 119868119873119879

120590

vanish








of ΨDO120590by

ΨDO120590119898

=

119872

sum

119894=119898

119886119894120585119894

| 119886119894isinA (61)




Γ119899= 120572 = (1205721 120572119894 120572119899) 120572119894 isinZ

Γ+

119899= 120572 isin Γ

119899 120572

119894isinZ

+

1le 119894 le 119899

120576119894= (0 0 1 0 0)

120599 =

119899

sum

119894=1120576119894isin Γ

119899

120572 =

119899

prod

119894=1120572119894 120572

119894isin Γ

+

119899

(62)


119899 120575120574(119887) = 12057512057411 ∘ sdot sdot sdot ∘ 120575

120574119899


1205751 120575119899 and

(120572

120574) = prod

119894

(120572119894

120574119894

) (63)


119894 with 119894 = 1 119899




120585119894119886 = 119886120585

119894+ 120575

119894(119886) (64)



119899


119886120585120572

sdot 119887120585120573

= sum

120574isinΓ+

119899

(120572

120574)119886120575

120574

(119887) 120585120572+120573minus120574

(65)


119899DOof formal


120585minus1119894119886 = sum

119895ge0(minus1)119895 120575119895

119894(119886) 120585


(66)


commutator

[119860 119861] = 119860119861minus119861119860

= sum

120574isinΓ+

119899

(1120574)(

120597120574

120597120585120574(119860) 120575

120574

(119861) minus120597120574

120597120585120574(119861) 120575

120574

(119860)) (67)


120597120574

120597120585120574(119886120585

120572

) = 120574 (120572

120574) 119886120585

120572minus120574

120575120574

(119886120585120572

) = 120575120574

(119886) 120585120572

(68)





119894 sdot]

considered in (67)

[log 120585119894 119883] = sum

120574isinΓ+

119899

1120574

120597120574

120597120585120574(log 120585

119894) 120575

120574

(119883)

= sum

120574119894ge0

1120574119894

120597120574

119894

120597120585120574119894

119894

(log 120585119894) 120575

120574119894 (119883)

(69)






[log 120585119894 119860119861] = [log 120585

119894 119860] 119861 +119860 [log 120585

119894 119861] (70)






119894(119886)) =


120591 (119886120575120574

(119887)) = (minus1)120574 120591 (119887120575120574 (119886)) (71)




res Ψ119899119863119874 rarr CΨ

119899119863119874 rarr C defined by


119886120572120585120572

) = 120591 (119886minus120599) (72)

is a trace on Ψ119899119863119874


res (119886120585120572119887120585120573) = res (119887120585120573119886120585120572) (73)


119894 120573

119895 we have 120572

119894 120573

119895ge 0

or 120572119894 120573



119894lt 0

If 120572119894+ 120573


Now suppose that 120572119894+ 120573


119894= 120572

119894+ 120573

119894 We have

res (119886120585120572119887120585120573) = (120572

119896 + 120599) 120591 (119886120575

119896+120599

) (74)


res (119887120585120573119886120585120572) = 120591((120573

119896 + 120599) 119887120575

119896+120599

(119886))

= (119896 minus 120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (minus1)119896+120599 (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

= (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

(75)





119894 119860]119861) 119894 =



119894 sdot] is


119894([119862 119860] 119861) +

119888119894([119861 119862] 119860) = 0




119894= 119898 gt 0 for fixed 1 le 119894 le

119899 and

Ψ119899DO

120572119894

=

sum

120573isinΓ119899

119887120573120585120573

= sum

(1205731 120573119899)isinΓ119899

11988612057312058512057311 sdot sdot sdot 120585

120573119899

119899

isinΨ119899DO 120573

119894ge120572

119894

(76)




120572119894


119899119863119874

120572119894




119899DOlowast

120572119894

≃ sum120579isinΓ119899

119897120579120585120579

120579119894le minus120572

119894minus 1 Let 119860 119861 isin

Ψ119899DO

120572119894



120572119894


120572119894

we have

119888 (119860 119861) = res ([119860 119861] 119871)

= res([

[

sum

120572isinΓ119899

119886120572120585120572

sum

120573isinΓ119899

119887120573120585120573]

]

sum

120579isinΓ119899

120579119894leminus120572119894minus1

119897120579120585120579

)


(])(119887120573) 120575

(120583)

(119897120579)

sdot 120585120572+120573+120579minus(]+120583+120593)


(77)


120572119894+120573

119894+ 120579

119894+ (]

119894+120583

119894+120593

119894) = minus 1 (78)


119888 (119860 119861) = res ([log 120585119894 119860] 119861)

= res([log 120585119894 sum

120572isinΓ119899

119886120572120585120572

] sum

120573isinΓ119899

119887120573120585120573

)


(119886120572) 120575

(120591119894)

(119887120573) 120585

120572+120573minus(120595119894+120591)

)


(79)


120572119894+120573

119894+ (120591

119894+120593

119894) = minus 1 (80)


119894= 120579


120579119894le minus120572

119894minus 1 and 120583

119894ge 0


119894


119899DO



120573isinΓ119899

119887120573120585120573 such that 120573 ge 120572






120572




119894= 120579



Let 119875+119899

= sum120572120582120572119909120572

| 120572 isin Γ+




120582120572119909120572 such that the





119894(119909

120572

) =

120572119894119909120572minus120598119894 with 120572 isin Γ

119899





120572isinΓ+

119899

1120572(119906120575

120572

(V) otimes 120575120572 (119891) 119892) (81)



have the following


119899119863119874 and


Ψ119899119863119874 997888rarr Uotimes119875

119899

119906120585120572

997891997888rarr 119906otimes119909120572

(82)


119899119863119874 andU otimes 119875

119899




+119909120573

minus≃ 119909

120572

otimes119909120573


sum

120572120573isinΓ119899

120582120572120573119909120572

+119909120573

minus (83)


[119909120572

+119909120573

minus 119909

120572

+119909120573

minus] = sum

120574isinΓ+

119899

(1120574) (120597

minusand 120597

+) (119909

120572

+119909120573

minus 119909

120572

+119909120573

minus) (84)


119899as

120597120574

minus(119909

120572

+119909120573

minus) = 120574 (

120573

120574)119909

120572

+119909120573minus120574

minus

120597120574

+(119909

120572

+119909120573

minus) = 120574 (

120572

120574)119909

120572minus120574

+119909120573

minus

(85)


+= 120597

minusotimes 120597

+minus 120597

+otimes 120597

minus


1198910(119909minus120599

) = 1 and 1198910(119909120572

) = 0 if 120572 = minus120599



this section



119899 1198671

(H119899H



minus119894 sdot] 119894 = 1 119899


+= 119909

+1 and 119909minus

= 119909minus1 We








119863 = 120582 [log119909+ sdot] + 120573 [log119909


for some 120582 120573


+) = sum119889

+

119899119898119909119899

+119909119898

minusand

119863(119909minus) = sum119889

minus

119899119898119909119899

+119909119898

minus We have

1 = [119909minus 119909

+] 997904rArr 120582 = [119863 (119909

minus) 119909

+] + [119909

minus 119863 (119909

+)]

997904rArr 120582 = sum119889minus

119899119898[119909

119899

+119909119898

minus 119909

+] +sum119889

+

119899119898[119909

minus 119909

119899

+119909119898

minus]

997904rArr 120582 = sum119889minus

119899119898119898119909

119899


+sum119889+

119899119898119899119909

119899minus1+119909119898

minus

(87)



+) = sum119889

+

0119898119909119898


that119863(119909minus) = sum119889

minus

1198990119909119899


119889minus

119899119898119898119909

119899


= minus119889119899119898119899119909

119899minus1+119909119898

minus(88)


(119898+ 1) 119889minus119899119898+1 = minus (119899 + 1) 119889119899+1119898 (89)


Then

119863(119909+) = sum

119898 =0minus1minus 119889

minus

0119898119909119898


minus1minus+119889

+

00 (90)

119863(119909minus) = sum

119899 =0minus1119889minus

1198990119909119899

++119889

minus10119909minus1++119889

minus

00 (91)

Now we write

119863(119909+) = 120582 [log119909

+ 119909

+] + 120573 [log119909

minus 119909

+] minus [120596 119909

+]


+]

(92)


[120596 119909+] = sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120597minus(120596)

= sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120596

= sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

+119889+

00119909minus +

(93)


119863(119909minus) = 120582 [log119909

+ 119909

minus] + 120573 [log119909

minus 119909

minus] minus [120596 119909

minus]

= 120582119909minus1+minus [120596 119909

minus]

(94)



119899 =0minus1119889minus

1198990119909119899

++minus119889

+

00 997904rArr 120597minus(120596)

= minus sum

119899 =0minus1119889minus

1198990119909119899

+minus119889

+

00 997904rArr 120596

= minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

minus119889minus

00119909+ +120596

(95)


120596 = minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

+ sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

minus119889minus

00119909+ minus119889minus

00119909minus

(96)









119894= res([log 120585

plusmn119894 119860]119861) and







4 Applications



+form a Manin







+and g



minus

vanish



120590 DO

120590 and


form ⟨119860 119861⟩120590



that 119886120572119899





INT119899= 119871 isinΨ

119899DO | ord (119871) lt 0

DO119899= 119871 isinΨ

119899DO | ord (119871) ge 0

(97)



119899oplus DO

119899


119899119863119874 119868119873119879

119899 119863119874








119899DO


INT119894=

(minus1)sum


119886120572120585120572

| 119886120572isin119860

DO119894=

(119899)

sum

(0)isinΓ119899

119886120572120585120572

| 119886120572isin119860

(98)


119894oplusDO

119894


119894 119863119874

119894) formaMa-









119891(A) where Der

119891(A)




(119886 119886) 997891997888rarr (119887 997891997888rarr119891 (120601 (119887) 119886 119886)) (99)



(119887 sdot 120573 ()) = minus120573 (119887) (100)



lowast

(119887 + 119886 +120573) ( + 119886 +120573)

= 119887 + 120601 (119887) 119886 minus 120601 () 119886 + 119886119886 +119908 (119886 119886) + 119887 sdot 120573

minus sdot 120573

(101)


onA119861

119891119861(119887 + 119886 +120573 + 119886 + 120573) = 120573 () + 120573 (119887) +119891 (119886 119886) (102)

Thepair (A119861 119891




119861follows


119908(119886 [1198861015840

119886]) = 119908 (1198861015840

[119886 119886]) minus119908 (119886 [119886 1198861015840

])

119887 sdot 119908 (1198861015840

119886) = 119908 (1198861015840

120601 (119887) sdot 119886) +119908 (120601 (119887) sdot 1198861015840

119886)

(103)





lowast with bracket

[(1198871 1198711 1205731) (1198872 1198712 1205732)] = (0 1198871 [log 120585 1198712]

minus 1198872 [log 120585 1198711] + [1198711 1198712] ⟨[log 120585 1198711] 1198712⟩)(104)


⟨(1198871 1198711 1205731) (1198872 1198712 1205732)⟩

= 12057311198872 +12057321198871 + res (11987111198712) (105)




120590)



120590oplus INT

120590) in which

DO = 120572119888 +

119899

sum

119894=0119886119894120585119894

| 119886119894isinA 120572 isinR


119894=minusinfin

119886119894120585119894

| 119886119894isinA 120573 isinR

(106)



(Ψ119863119874119863119874 119868119873119879)

(Ψ120590119863119874119863119874

120590 119868119873119879

120590)

(107)

are Manin triples




119899DO = Ψ


119894



Ψ119899DO = INT

119894oplus DO

119894 (108)

where

INT119894=

120573 log 120585119894+

(minus1)sum


119886120579120585120579

| 119886120579isin119860

DO119894=

120572119888119894+

(119899)

sum

(0)isinΓ119899

119886120579120585120579

| 119886120579isin119860

(109)



119894 119863119874

119894) form a Ma-






decomposition

Ψ119899DO = Ψ




[(119860 120572 120573) (119861 120572 120573)] = ([119860 119861] + [120573 sdot log 120585 119860]

minus [120573 sdot log 120585 119861] 119888 (119860 119861) 0) (111)

and bilinear form

⟨119860 119861⟩ = res (119860 119861) + 120572 sdot 120573 +120573 sdot 120572 (112)


119899) 0 = (0 0) isin R119899 and 119888(119860 119861) = (1198881(119860 119861)





⟨]120575119865

120575120583⟩ =

119889

119889120598

10038161003816100381610038161003816100381610038161003816120598=0119865 (120583 + 120598]) (113)



119865 119866 (120583) = ⟨120583 [120575119865

120575120583120575119866

120575120583]⟩ (114)



119889120583

119889119905= ⟨120583 [sdot

120575119867

120575120583]⟩ (115)


119877on g by

[119875 119876]119877= [119877 (119875) 119876] + [119875 119877 (119876)] 119876 119875 isin g (116)





+minus



plusmn We write

119875plusmn= 120587


119877= [119875

+ 119876

+]minus[119875

minus 119876

minus] and

119865 119866119877= ⟨120583 [

120575119865

120575120583120575119866

120575120583]

119877

⟩ (117)


⟨120583 [120575119865

120575120583 sdot]⟩ = 0 (118)


119865 119866119877= ⟨120583 [(

120575119865

120575120583)

+

120575119866

120575120583]⟩ (119)


119883119867(120583) sdot 119865 = ⟨120583 [minus(

120575119867

120575120583)

+

120575119865

120575120583]⟩ (120)


⟨119875 [119876 119877]⟩ = ⟨[119877 119875] 119876⟩ (121)


119889119875

119889119905= [(

120575119867

120575120583)

+

119875] (122)



⟨[119875 119876] 119878⟩ = ⟨[119878 119875] 119876⟩ 119878 119875 119876 isin ΨDO (123)


minusoplus ΨDO

+allows us to




Ψ119863119874 Then

120575119867119896

120575119871= 119896119871

119896minus1 (124)


119867119896 119867

119897119877(119871) = 0 for any 119877-matrix 119877




119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (125)


120590 DO

120590 and INT

120590


120590as



120590 The


119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (126)



119899DO DO

119899 and INT

119899allows us


[ ]119877 Once more we set 119867

119896(119871) = res(119871119896) for 119871 isin Ψ

119899DO


120597119871

120597119905= [(119896119871

119896minus1)+

119871] (127)




⟨(119860 120572) (119861 120573)⟩ = ⟨119860 119861⟩ + 120572120573 = res (119860119861) + 120572120573 (128)




⟨120597119871

120597119905 119876⟩ = ⟨[

120597119867

120597119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (129)

119889119897

119889119905= 0 (130)



119889 ( )

119889119905sdot (119876 119903) = ⟨( ) [(119876 119903)

120575119867

120575 ( )]⟩

= ⟨( ) [(119876 119903) (120575119867

120575120575119867

120575)]⟩

= ⟨( ) ([119876120575119867

120575] 119888 (119876

120575119867

120575))⟩

(131)


⟨120597 (119871 119897)

120597119905 (119876 119903)⟩

= ⟨(119871 119897) ([119876120575119867

120575119871] 119888 (119876

120575119867

120575119871))⟩

= ⟨119871 [119876120575119867

120575119871]⟩+ 119897119888 (119876

120575119867

120575119871)

(132)


⟨120597119871

120597119905 119876⟩+ 119903

120597119897

120597119905= ⟨[

120575119867

120575119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (133)





res ([log 120585 119886120585] 119887120585) = res (119887120585 [log 120585 119886120585])

= res(119887120585infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 1205851minus119895)

= res(infin

sum

119895=0

(minus1)119895+1

119895(119887120575

119895

(119886) 1205852minus119895

+ 119887120575119895+1

(119886) 1205851minus119895))

=(minus1)3+1

3119887120575

3(119886) +

(minus1)2+1

2119887120575

3(119886)

= minus16120591 (119887120575

3(119886))

(134)


⟨(119886120585 120572) (119887120585 120573)⟩ = 120591 (119886119887) + 120572120573 (135)


⟨119889119906

119889119905120585 119887120585⟩ = ⟨[ℎ (119906) 120585 119906120585] 119887120585⟩

minus16119897120591 (119887120575

3(ℎ (119906)))

(136)


119889119906

119889119905= ℎ (119906) 120575 (119906) minus 119906120575 (ℎ (119906)) minus (

16) 119897120575

3(ℎ (119906)) (137)




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119888 (119891 (119909) 120597 119892 (119909) 120597) =16int119891

1015840

(119909) 11989210158401015840

(119909) 119889119909 (40)



119888 (119891120597119899

119892120597119898

) =119899119898

(119898 + 119899 + 1)int1198781119891(119898+1)

119892(119899)

119889120579 (41)


119888 (119860 119861) = int1198781([119909 119860] ∘ 119861) (42)



1015840


119872ΨDO =

119899

sum

119894=minusinfin

119886119894119863V

119894

| 119886119894isinM (43)


119863(sum

119899

119894=minusinfin119886119894119863119894

V) = 119886minus1119863



119875(res

119863(119860))Then as










119898≃ sum


119886119894120585119894



119894=119898119886119894120585119894 119861 =

sum119872

119894=119898119887119894120585119894



119897119894120585119894

isin ΨDOlowast

119898we

have

119888 (119860 119861) = res ([119860 119861] 119871) = res([

[

119872

sum

119894=119898

119886119894120585119894

119872

sum

119895=119898

119887119895120585119895]

]

sdot


119896=minusinfin

119897119896120585119896

)

= res( sum


119894)

sdot 120575(119905)

(119887119895) 120575

(119906)

(119897119896) 120585

119894+119895+119896minus(119905+119906+119904)

))


(44)


119894 + 119895 + 119896 minus (119905 + 119906 + 119904) = minus 1 (45)


119888 (119860 119861) = res ([log 120585 119860] 119861) = res([log 120585119872

sum

119894=119898

119886119894120585119894

]

sdot

119872

sum

119895=119898

119887119895120585119895

) = res( sum

119894119895119905119904=11989811989800(coefficient) 120575(119904) (119886

119894)

sdot 120575(119905)

(119887119895) 120585

119894+119895minus(119904+119905)


(46)


119894 + 119895 minus (119904 + 119905) = minus 1 (47)



119898









ΨDO120590=

119873

sum

119894=minusinfin

119886119894120585119894

119873 isinZ 119886119899isinA (48)


120585119886 = 120590 (119886) 120585 + 120575 (119886)

120585120585minus1= 120585

minus1120585 = 1

(49)


120585119899

119886 =

119899

sum

119894=0119875119894119899(120590 120575) (119886) 120585

119894

(50)





120585119899

119886 =

infin

sum


infin

sum


sdot 120590minus1(120575120590

minus1)119894119899


minus1)1198941(119886) 120585


(51)



119898=minusinfin119887119898120585119898

then

119860119861 =

119872

sum

119898=minusinfin

sum

119899lt0sum

119894ge0(minus1)|119894| 119886

119899120590minus1(120575120590



minus1)1198941(119887119898) 120585

119898+119899+|119894|

+

119872

sum

119898=minusinfin

119873

sum

119899=0

119899

sum

119895=0119886119899119875119895119899(120590 120575) (119887

119898) 120585

119898+119895

(52)




120591 (119887120590minus1(120575120590


minus1(120575120590

minus1)119894119898

(119886))




res(119899

sum

119894=minusinfin

119886119894120585119894

) = 120591 (119886minus1) (54)

is a trace on Ψ119863119874120590


120585119899

119886 =

infin

sum

119895=0(119899

119895) 120575

119895

(120590119899minus119895

(119886)) 120585119899minus119895

(55)


119905be a 1-


120585119905

119886 minus 120590119905(119886) 120585

119905

=

infin

sum

119895=1(119905

119895) 120575

119895

(120590119905minus119895) 120585

119905minus119895

(56)


[log 120585 119886] = 119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120590119905(119886) +

infin

sum

119895=1


119895120590minus119895

120575119895

(119886) 120585minus119895

(57)



rarr Ψ119863119874120590

defined by

[log 120585 119886120585119899] = 119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120590119905(119886) 120585

119899

+

infin

sum

119895=1


119895120590minus119895

120575119895

(119886) 120585minus119895

(58)

is a derivation




ΨDO120590= DO

120590oplus INT

120590 (59)

where

DO120590=

119899

sum

119894=0119886119894120585119894

| 119886119894isinA

INT120590=

minus1sum

119894=minusinfin

119886119894120585119894

| 119886119894isinA

(60)







algebras 119863119874120590and 119868119873119879



120590and 119868119873119879

120590

vanish








of ΨDO120590by

ΨDO120590119898

=

119872

sum

119894=119898

119886119894120585119894

| 119886119894isinA (61)




Γ119899= 120572 = (1205721 120572119894 120572119899) 120572119894 isinZ

Γ+

119899= 120572 isin Γ

119899 120572

119894isinZ

+

1le 119894 le 119899

120576119894= (0 0 1 0 0)

120599 =

119899

sum

119894=1120576119894isin Γ

119899

120572 =

119899

prod

119894=1120572119894 120572

119894isin Γ

+

119899

(62)


119899 120575120574(119887) = 12057512057411 ∘ sdot sdot sdot ∘ 120575

120574119899


1205751 120575119899 and

(120572

120574) = prod

119894

(120572119894

120574119894

) (63)


119894 with 119894 = 1 119899




120585119894119886 = 119886120585

119894+ 120575

119894(119886) (64)



119899


119886120585120572

sdot 119887120585120573

= sum

120574isinΓ+

119899

(120572

120574)119886120575

120574

(119887) 120585120572+120573minus120574

(65)


119899DOof formal


120585minus1119894119886 = sum

119895ge0(minus1)119895 120575119895

119894(119886) 120585


(66)


commutator

[119860 119861] = 119860119861minus119861119860

= sum

120574isinΓ+

119899

(1120574)(

120597120574

120597120585120574(119860) 120575

120574

(119861) minus120597120574

120597120585120574(119861) 120575

120574

(119860)) (67)


120597120574

120597120585120574(119886120585

120572

) = 120574 (120572

120574) 119886120585

120572minus120574

120575120574

(119886120585120572

) = 120575120574

(119886) 120585120572

(68)





119894 sdot]

considered in (67)

[log 120585119894 119883] = sum

120574isinΓ+

119899

1120574

120597120574

120597120585120574(log 120585

119894) 120575

120574

(119883)

= sum

120574119894ge0

1120574119894

120597120574

119894

120597120585120574119894

119894

(log 120585119894) 120575

120574119894 (119883)

(69)






[log 120585119894 119860119861] = [log 120585

119894 119860] 119861 +119860 [log 120585

119894 119861] (70)






119894(119886)) =


120591 (119886120575120574

(119887)) = (minus1)120574 120591 (119887120575120574 (119886)) (71)




res Ψ119899119863119874 rarr CΨ

119899119863119874 rarr C defined by


119886120572120585120572

) = 120591 (119886minus120599) (72)

is a trace on Ψ119899119863119874


res (119886120585120572119887120585120573) = res (119887120585120573119886120585120572) (73)


119894 120573

119895 we have 120572

119894 120573

119895ge 0

or 120572119894 120573



119894lt 0

If 120572119894+ 120573


Now suppose that 120572119894+ 120573


119894= 120572

119894+ 120573

119894 We have

res (119886120585120572119887120585120573) = (120572

119896 + 120599) 120591 (119886120575

119896+120599

) (74)


res (119887120585120573119886120585120572) = 120591((120573

119896 + 120599) 119887120575

119896+120599

(119886))

= (119896 minus 120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (minus1)119896+120599 (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

= (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

(75)





119894 119860]119861) 119894 =



119894 sdot] is


119894([119862 119860] 119861) +

119888119894([119861 119862] 119860) = 0




119894= 119898 gt 0 for fixed 1 le 119894 le

119899 and

Ψ119899DO

120572119894

=

sum

120573isinΓ119899

119887120573120585120573

= sum

(1205731 120573119899)isinΓ119899

11988612057312058512057311 sdot sdot sdot 120585

120573119899

119899

isinΨ119899DO 120573

119894ge120572

119894

(76)




120572119894


119899119863119874

120572119894




119899DOlowast

120572119894

≃ sum120579isinΓ119899

119897120579120585120579

120579119894le minus120572

119894minus 1 Let 119860 119861 isin

Ψ119899DO

120572119894



120572119894


120572119894

we have

119888 (119860 119861) = res ([119860 119861] 119871)

= res([

[

sum

120572isinΓ119899

119886120572120585120572

sum

120573isinΓ119899

119887120573120585120573]

]

sum

120579isinΓ119899

120579119894leminus120572119894minus1

119897120579120585120579

)


(])(119887120573) 120575

(120583)

(119897120579)

sdot 120585120572+120573+120579minus(]+120583+120593)


(77)


120572119894+120573

119894+ 120579

119894+ (]

119894+120583

119894+120593

119894) = minus 1 (78)


119888 (119860 119861) = res ([log 120585119894 119860] 119861)

= res([log 120585119894 sum

120572isinΓ119899

119886120572120585120572

] sum

120573isinΓ119899

119887120573120585120573

)


(119886120572) 120575

(120591119894)

(119887120573) 120585

120572+120573minus(120595119894+120591)

)


(79)


120572119894+120573

119894+ (120591

119894+120593

119894) = minus 1 (80)


119894= 120579


120579119894le minus120572

119894minus 1 and 120583

119894ge 0


119894


119899DO



120573isinΓ119899

119887120573120585120573 such that 120573 ge 120572






120572




119894= 120579



Let 119875+119899

= sum120572120582120572119909120572

| 120572 isin Γ+




120582120572119909120572 such that the





119894(119909

120572

) =

120572119894119909120572minus120598119894 with 120572 isin Γ

119899





120572isinΓ+

119899

1120572(119906120575

120572

(V) otimes 120575120572 (119891) 119892) (81)



have the following


119899119863119874 and


Ψ119899119863119874 997888rarr Uotimes119875

119899

119906120585120572

997891997888rarr 119906otimes119909120572

(82)


119899119863119874 andU otimes 119875

119899




+119909120573

minus≃ 119909

120572

otimes119909120573


sum

120572120573isinΓ119899

120582120572120573119909120572

+119909120573

minus (83)


[119909120572

+119909120573

minus 119909

120572

+119909120573

minus] = sum

120574isinΓ+

119899

(1120574) (120597

minusand 120597

+) (119909

120572

+119909120573

minus 119909

120572

+119909120573

minus) (84)


119899as

120597120574

minus(119909

120572

+119909120573

minus) = 120574 (

120573

120574)119909

120572

+119909120573minus120574

minus

120597120574

+(119909

120572

+119909120573

minus) = 120574 (

120572

120574)119909

120572minus120574

+119909120573

minus

(85)


+= 120597

minusotimes 120597

+minus 120597

+otimes 120597

minus


1198910(119909minus120599

) = 1 and 1198910(119909120572

) = 0 if 120572 = minus120599



this section



119899 1198671

(H119899H



minus119894 sdot] 119894 = 1 119899


+= 119909

+1 and 119909minus

= 119909minus1 We








119863 = 120582 [log119909+ sdot] + 120573 [log119909


for some 120582 120573


+) = sum119889

+

119899119898119909119899

+119909119898

minusand

119863(119909minus) = sum119889

minus

119899119898119909119899

+119909119898

minus We have

1 = [119909minus 119909

+] 997904rArr 120582 = [119863 (119909

minus) 119909

+] + [119909

minus 119863 (119909

+)]

997904rArr 120582 = sum119889minus

119899119898[119909

119899

+119909119898

minus 119909

+] +sum119889

+

119899119898[119909

minus 119909

119899

+119909119898

minus]

997904rArr 120582 = sum119889minus

119899119898119898119909

119899


+sum119889+

119899119898119899119909

119899minus1+119909119898

minus

(87)



+) = sum119889

+

0119898119909119898


that119863(119909minus) = sum119889

minus

1198990119909119899


119889minus

119899119898119898119909

119899


= minus119889119899119898119899119909

119899minus1+119909119898

minus(88)


(119898+ 1) 119889minus119899119898+1 = minus (119899 + 1) 119889119899+1119898 (89)


Then

119863(119909+) = sum

119898 =0minus1minus 119889

minus

0119898119909119898


minus1minus+119889

+

00 (90)

119863(119909minus) = sum

119899 =0minus1119889minus

1198990119909119899

++119889

minus10119909minus1++119889

minus

00 (91)

Now we write

119863(119909+) = 120582 [log119909

+ 119909

+] + 120573 [log119909

minus 119909

+] minus [120596 119909

+]


+]

(92)


[120596 119909+] = sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120597minus(120596)

= sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120596

= sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

+119889+

00119909minus +

(93)


119863(119909minus) = 120582 [log119909

+ 119909

minus] + 120573 [log119909

minus 119909

minus] minus [120596 119909

minus]

= 120582119909minus1+minus [120596 119909

minus]

(94)



119899 =0minus1119889minus

1198990119909119899

++minus119889

+

00 997904rArr 120597minus(120596)

= minus sum

119899 =0minus1119889minus

1198990119909119899

+minus119889

+

00 997904rArr 120596

= minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

minus119889minus

00119909+ +120596

(95)


120596 = minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

+ sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

minus119889minus

00119909+ minus119889minus

00119909minus

(96)









119894= res([log 120585

plusmn119894 119860]119861) and







4 Applications



+form a Manin







+and g



minus

vanish



120590 DO

120590 and


form ⟨119860 119861⟩120590



that 119886120572119899





INT119899= 119871 isinΨ

119899DO | ord (119871) lt 0

DO119899= 119871 isinΨ

119899DO | ord (119871) ge 0

(97)



119899oplus DO

119899


119899119863119874 119868119873119879

119899 119863119874








119899DO


INT119894=

(minus1)sum


119886120572120585120572

| 119886120572isin119860

DO119894=

(119899)

sum

(0)isinΓ119899

119886120572120585120572

| 119886120572isin119860

(98)


119894oplusDO

119894


119894 119863119874

119894) formaMa-









119891(A) where Der

119891(A)




(119886 119886) 997891997888rarr (119887 997891997888rarr119891 (120601 (119887) 119886 119886)) (99)



(119887 sdot 120573 ()) = minus120573 (119887) (100)



lowast

(119887 + 119886 +120573) ( + 119886 +120573)

= 119887 + 120601 (119887) 119886 minus 120601 () 119886 + 119886119886 +119908 (119886 119886) + 119887 sdot 120573

minus sdot 120573

(101)


onA119861

119891119861(119887 + 119886 +120573 + 119886 + 120573) = 120573 () + 120573 (119887) +119891 (119886 119886) (102)

Thepair (A119861 119891




119861follows


119908(119886 [1198861015840

119886]) = 119908 (1198861015840

[119886 119886]) minus119908 (119886 [119886 1198861015840

])

119887 sdot 119908 (1198861015840

119886) = 119908 (1198861015840

120601 (119887) sdot 119886) +119908 (120601 (119887) sdot 1198861015840

119886)

(103)





lowast with bracket

[(1198871 1198711 1205731) (1198872 1198712 1205732)] = (0 1198871 [log 120585 1198712]

minus 1198872 [log 120585 1198711] + [1198711 1198712] ⟨[log 120585 1198711] 1198712⟩)(104)


⟨(1198871 1198711 1205731) (1198872 1198712 1205732)⟩

= 12057311198872 +12057321198871 + res (11987111198712) (105)




120590)



120590oplus INT

120590) in which

DO = 120572119888 +

119899

sum

119894=0119886119894120585119894

| 119886119894isinA 120572 isinR


119894=minusinfin

119886119894120585119894

| 119886119894isinA 120573 isinR

(106)



(Ψ119863119874119863119874 119868119873119879)

(Ψ120590119863119874119863119874

120590 119868119873119879

120590)

(107)

are Manin triples




119899DO = Ψ


119894



Ψ119899DO = INT

119894oplus DO

119894 (108)

where

INT119894=

120573 log 120585119894+

(minus1)sum


119886120579120585120579

| 119886120579isin119860

DO119894=

120572119888119894+

(119899)

sum

(0)isinΓ119899

119886120579120585120579

| 119886120579isin119860

(109)



119894 119863119874

119894) form a Ma-






decomposition

Ψ119899DO = Ψ




[(119860 120572 120573) (119861 120572 120573)] = ([119860 119861] + [120573 sdot log 120585 119860]

minus [120573 sdot log 120585 119861] 119888 (119860 119861) 0) (111)

and bilinear form

⟨119860 119861⟩ = res (119860 119861) + 120572 sdot 120573 +120573 sdot 120572 (112)


119899) 0 = (0 0) isin R119899 and 119888(119860 119861) = (1198881(119860 119861)





⟨]120575119865

120575120583⟩ =

119889

119889120598

10038161003816100381610038161003816100381610038161003816120598=0119865 (120583 + 120598]) (113)



119865 119866 (120583) = ⟨120583 [120575119865

120575120583120575119866

120575120583]⟩ (114)



119889120583

119889119905= ⟨120583 [sdot

120575119867

120575120583]⟩ (115)


119877on g by

[119875 119876]119877= [119877 (119875) 119876] + [119875 119877 (119876)] 119876 119875 isin g (116)





+minus



plusmn We write

119875plusmn= 120587


119877= [119875

+ 119876

+]minus[119875

minus 119876

minus] and

119865 119866119877= ⟨120583 [

120575119865

120575120583120575119866

120575120583]

119877

⟩ (117)


⟨120583 [120575119865

120575120583 sdot]⟩ = 0 (118)


119865 119866119877= ⟨120583 [(

120575119865

120575120583)

+

120575119866

120575120583]⟩ (119)


119883119867(120583) sdot 119865 = ⟨120583 [minus(

120575119867

120575120583)

+

120575119865

120575120583]⟩ (120)


⟨119875 [119876 119877]⟩ = ⟨[119877 119875] 119876⟩ (121)


119889119875

119889119905= [(

120575119867

120575120583)

+

119875] (122)



⟨[119875 119876] 119878⟩ = ⟨[119878 119875] 119876⟩ 119878 119875 119876 isin ΨDO (123)


minusoplus ΨDO

+allows us to




Ψ119863119874 Then

120575119867119896

120575119871= 119896119871

119896minus1 (124)


119867119896 119867

119897119877(119871) = 0 for any 119877-matrix 119877




119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (125)


120590 DO

120590 and INT

120590


120590as



120590 The


119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (126)



119899DO DO

119899 and INT

119899allows us


[ ]119877 Once more we set 119867

119896(119871) = res(119871119896) for 119871 isin Ψ

119899DO


120597119871

120597119905= [(119896119871

119896minus1)+

119871] (127)




⟨(119860 120572) (119861 120573)⟩ = ⟨119860 119861⟩ + 120572120573 = res (119860119861) + 120572120573 (128)




⟨120597119871

120597119905 119876⟩ = ⟨[

120597119867

120597119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (129)

119889119897

119889119905= 0 (130)



119889 ( )

119889119905sdot (119876 119903) = ⟨( ) [(119876 119903)

120575119867

120575 ( )]⟩

= ⟨( ) [(119876 119903) (120575119867

120575120575119867

120575)]⟩

= ⟨( ) ([119876120575119867

120575] 119888 (119876

120575119867

120575))⟩

(131)


⟨120597 (119871 119897)

120597119905 (119876 119903)⟩

= ⟨(119871 119897) ([119876120575119867

120575119871] 119888 (119876

120575119867

120575119871))⟩

= ⟨119871 [119876120575119867

120575119871]⟩+ 119897119888 (119876

120575119867

120575119871)

(132)


⟨120597119871

120597119905 119876⟩+ 119903

120597119897

120597119905= ⟨[

120575119867

120575119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (133)





res ([log 120585 119886120585] 119887120585) = res (119887120585 [log 120585 119886120585])

= res(119887120585infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 1205851minus119895)

= res(infin

sum

119895=0

(minus1)119895+1

119895(119887120575

119895

(119886) 1205852minus119895

+ 119887120575119895+1

(119886) 1205851minus119895))

=(minus1)3+1

3119887120575

3(119886) +

(minus1)2+1

2119887120575

3(119886)

= minus16120591 (119887120575

3(119886))

(134)


⟨(119886120585 120572) (119887120585 120573)⟩ = 120591 (119886119887) + 120572120573 (135)


⟨119889119906

119889119905120585 119887120585⟩ = ⟨[ℎ (119906) 120585 119906120585] 119887120585⟩

minus16119897120591 (119887120575

3(ℎ (119906)))

(136)


119889119906

119889119905= ℎ (119906) 120575 (119906) minus 119906120575 (ℎ (119906)) minus (

16) 119897120575

3(ℎ (119906)) (137)




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















ΨDO120590=

119873

sum

119894=minusinfin

119886119894120585119894

119873 isinZ 119886119899isinA (48)


120585119886 = 120590 (119886) 120585 + 120575 (119886)

120585120585minus1= 120585

minus1120585 = 1

(49)


120585119899

119886 =

119899

sum

119894=0119875119894119899(120590 120575) (119886) 120585

119894

(50)





120585119899

119886 =

infin

sum


infin

sum


sdot 120590minus1(120575120590

minus1)119894119899


minus1)1198941(119886) 120585


(51)



119898=minusinfin119887119898120585119898

then

119860119861 =

119872

sum

119898=minusinfin

sum

119899lt0sum

119894ge0(minus1)|119894| 119886

119899120590minus1(120575120590



minus1)1198941(119887119898) 120585

119898+119899+|119894|

+

119872

sum

119898=minusinfin

119873

sum

119899=0

119899

sum

119895=0119886119899119875119895119899(120590 120575) (119887

119898) 120585

119898+119895

(52)




120591 (119887120590minus1(120575120590


minus1(120575120590

minus1)119894119898

(119886))




res(119899

sum

119894=minusinfin

119886119894120585119894

) = 120591 (119886minus1) (54)

is a trace on Ψ119863119874120590


120585119899

119886 =

infin

sum

119895=0(119899

119895) 120575

119895

(120590119899minus119895

(119886)) 120585119899minus119895

(55)


119905be a 1-


120585119905

119886 minus 120590119905(119886) 120585

119905

=

infin

sum

119895=1(119905

119895) 120575

119895

(120590119905minus119895) 120585

119905minus119895

(56)


[log 120585 119886] = 119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120590119905(119886) +

infin

sum

119895=1


119895120590minus119895

120575119895

(119886) 120585minus119895

(57)



rarr Ψ119863119874120590

defined by

[log 120585 119886120585119899] = 119889

119889119905

10038161003816100381610038161003816100381610038161003816119905=0120590119905(119886) 120585

119899

+

infin

sum

119895=1


119895120590minus119895

120575119895

(119886) 120585minus119895

(58)

is a derivation




ΨDO120590= DO

120590oplus INT

120590 (59)

where

DO120590=

119899

sum

119894=0119886119894120585119894

| 119886119894isinA

INT120590=

minus1sum

119894=minusinfin

119886119894120585119894

| 119886119894isinA

(60)







algebras 119863119874120590and 119868119873119879



120590and 119868119873119879

120590

vanish








of ΨDO120590by

ΨDO120590119898

=

119872

sum

119894=119898

119886119894120585119894

| 119886119894isinA (61)




Γ119899= 120572 = (1205721 120572119894 120572119899) 120572119894 isinZ

Γ+

119899= 120572 isin Γ

119899 120572

119894isinZ

+

1le 119894 le 119899

120576119894= (0 0 1 0 0)

120599 =

119899

sum

119894=1120576119894isin Γ

119899

120572 =

119899

prod

119894=1120572119894 120572

119894isin Γ

+

119899

(62)


119899 120575120574(119887) = 12057512057411 ∘ sdot sdot sdot ∘ 120575

120574119899


1205751 120575119899 and

(120572

120574) = prod

119894

(120572119894

120574119894

) (63)


119894 with 119894 = 1 119899




120585119894119886 = 119886120585

119894+ 120575

119894(119886) (64)



119899


119886120585120572

sdot 119887120585120573

= sum

120574isinΓ+

119899

(120572

120574)119886120575

120574

(119887) 120585120572+120573minus120574

(65)


119899DOof formal


120585minus1119894119886 = sum

119895ge0(minus1)119895 120575119895

119894(119886) 120585


(66)


commutator

[119860 119861] = 119860119861minus119861119860

= sum

120574isinΓ+

119899

(1120574)(

120597120574

120597120585120574(119860) 120575

120574

(119861) minus120597120574

120597120585120574(119861) 120575

120574

(119860)) (67)


120597120574

120597120585120574(119886120585

120572

) = 120574 (120572

120574) 119886120585

120572minus120574

120575120574

(119886120585120572

) = 120575120574

(119886) 120585120572

(68)





119894 sdot]

considered in (67)

[log 120585119894 119883] = sum

120574isinΓ+

119899

1120574

120597120574

120597120585120574(log 120585

119894) 120575

120574

(119883)

= sum

120574119894ge0

1120574119894

120597120574

119894

120597120585120574119894

119894

(log 120585119894) 120575

120574119894 (119883)

(69)






[log 120585119894 119860119861] = [log 120585

119894 119860] 119861 +119860 [log 120585

119894 119861] (70)






119894(119886)) =


120591 (119886120575120574

(119887)) = (minus1)120574 120591 (119887120575120574 (119886)) (71)




res Ψ119899119863119874 rarr CΨ

119899119863119874 rarr C defined by


119886120572120585120572

) = 120591 (119886minus120599) (72)

is a trace on Ψ119899119863119874


res (119886120585120572119887120585120573) = res (119887120585120573119886120585120572) (73)


119894 120573

119895 we have 120572

119894 120573

119895ge 0

or 120572119894 120573



119894lt 0

If 120572119894+ 120573


Now suppose that 120572119894+ 120573


119894= 120572

119894+ 120573

119894 We have

res (119886120585120572119887120585120573) = (120572

119896 + 120599) 120591 (119886120575

119896+120599

) (74)


res (119887120585120573119886120585120572) = 120591((120573

119896 + 120599) 119887120575

119896+120599

(119886))

= (119896 minus 120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (minus1)119896+120599 (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

= (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

(75)





119894 119860]119861) 119894 =



119894 sdot] is


119894([119862 119860] 119861) +

119888119894([119861 119862] 119860) = 0




119894= 119898 gt 0 for fixed 1 le 119894 le

119899 and

Ψ119899DO

120572119894

=

sum

120573isinΓ119899

119887120573120585120573

= sum

(1205731 120573119899)isinΓ119899

11988612057312058512057311 sdot sdot sdot 120585

120573119899

119899

isinΨ119899DO 120573

119894ge120572

119894

(76)




120572119894


119899119863119874

120572119894




119899DOlowast

120572119894

≃ sum120579isinΓ119899

119897120579120585120579

120579119894le minus120572

119894minus 1 Let 119860 119861 isin

Ψ119899DO

120572119894



120572119894


120572119894

we have

119888 (119860 119861) = res ([119860 119861] 119871)

= res([

[

sum

120572isinΓ119899

119886120572120585120572

sum

120573isinΓ119899

119887120573120585120573]

]

sum

120579isinΓ119899

120579119894leminus120572119894minus1

119897120579120585120579

)


(])(119887120573) 120575

(120583)

(119897120579)

sdot 120585120572+120573+120579minus(]+120583+120593)


(77)


120572119894+120573

119894+ 120579

119894+ (]

119894+120583

119894+120593

119894) = minus 1 (78)


119888 (119860 119861) = res ([log 120585119894 119860] 119861)

= res([log 120585119894 sum

120572isinΓ119899

119886120572120585120572

] sum

120573isinΓ119899

119887120573120585120573

)


(119886120572) 120575

(120591119894)

(119887120573) 120585

120572+120573minus(120595119894+120591)

)


(79)


120572119894+120573

119894+ (120591

119894+120593

119894) = minus 1 (80)


119894= 120579


120579119894le minus120572

119894minus 1 and 120583

119894ge 0


119894


119899DO



120573isinΓ119899

119887120573120585120573 such that 120573 ge 120572






120572




119894= 120579



Let 119875+119899

= sum120572120582120572119909120572

| 120572 isin Γ+




120582120572119909120572 such that the





119894(119909

120572

) =

120572119894119909120572minus120598119894 with 120572 isin Γ

119899





120572isinΓ+

119899

1120572(119906120575

120572

(V) otimes 120575120572 (119891) 119892) (81)



have the following


119899119863119874 and


Ψ119899119863119874 997888rarr Uotimes119875

119899

119906120585120572

997891997888rarr 119906otimes119909120572

(82)


119899119863119874 andU otimes 119875

119899




+119909120573

minus≃ 119909

120572

otimes119909120573


sum

120572120573isinΓ119899

120582120572120573119909120572

+119909120573

minus (83)


[119909120572

+119909120573

minus 119909

120572

+119909120573

minus] = sum

120574isinΓ+

119899

(1120574) (120597

minusand 120597

+) (119909

120572

+119909120573

minus 119909

120572

+119909120573

minus) (84)


119899as

120597120574

minus(119909

120572

+119909120573

minus) = 120574 (

120573

120574)119909

120572

+119909120573minus120574

minus

120597120574

+(119909

120572

+119909120573

minus) = 120574 (

120572

120574)119909

120572minus120574

+119909120573

minus

(85)


+= 120597

minusotimes 120597

+minus 120597

+otimes 120597

minus


1198910(119909minus120599

) = 1 and 1198910(119909120572

) = 0 if 120572 = minus120599



this section



119899 1198671

(H119899H



minus119894 sdot] 119894 = 1 119899


+= 119909

+1 and 119909minus

= 119909minus1 We








119863 = 120582 [log119909+ sdot] + 120573 [log119909


for some 120582 120573


+) = sum119889

+

119899119898119909119899

+119909119898

minusand

119863(119909minus) = sum119889

minus

119899119898119909119899

+119909119898

minus We have

1 = [119909minus 119909

+] 997904rArr 120582 = [119863 (119909

minus) 119909

+] + [119909

minus 119863 (119909

+)]

997904rArr 120582 = sum119889minus

119899119898[119909

119899

+119909119898

minus 119909

+] +sum119889

+

119899119898[119909

minus 119909

119899

+119909119898

minus]

997904rArr 120582 = sum119889minus

119899119898119898119909

119899


+sum119889+

119899119898119899119909

119899minus1+119909119898

minus

(87)



+) = sum119889

+

0119898119909119898


that119863(119909minus) = sum119889

minus

1198990119909119899


119889minus

119899119898119898119909

119899


= minus119889119899119898119899119909

119899minus1+119909119898

minus(88)


(119898+ 1) 119889minus119899119898+1 = minus (119899 + 1) 119889119899+1119898 (89)


Then

119863(119909+) = sum

119898 =0minus1minus 119889

minus

0119898119909119898


minus1minus+119889

+

00 (90)

119863(119909minus) = sum

119899 =0minus1119889minus

1198990119909119899

++119889

minus10119909minus1++119889

minus

00 (91)

Now we write

119863(119909+) = 120582 [log119909

+ 119909

+] + 120573 [log119909

minus 119909

+] minus [120596 119909

+]


+]

(92)


[120596 119909+] = sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120597minus(120596)

= sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120596

= sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

+119889+

00119909minus +

(93)


119863(119909minus) = 120582 [log119909

+ 119909

minus] + 120573 [log119909

minus 119909

minus] minus [120596 119909

minus]

= 120582119909minus1+minus [120596 119909

minus]

(94)



119899 =0minus1119889minus

1198990119909119899

++minus119889

+

00 997904rArr 120597minus(120596)

= minus sum

119899 =0minus1119889minus

1198990119909119899

+minus119889

+

00 997904rArr 120596

= minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

minus119889minus

00119909+ +120596

(95)


120596 = minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

+ sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

minus119889minus

00119909+ minus119889minus

00119909minus

(96)









119894= res([log 120585

plusmn119894 119860]119861) and







4 Applications



+form a Manin







+and g



minus

vanish



120590 DO

120590 and


form ⟨119860 119861⟩120590



that 119886120572119899





INT119899= 119871 isinΨ

119899DO | ord (119871) lt 0

DO119899= 119871 isinΨ

119899DO | ord (119871) ge 0

(97)



119899oplus DO

119899


119899119863119874 119868119873119879

119899 119863119874








119899DO


INT119894=

(minus1)sum


119886120572120585120572

| 119886120572isin119860

DO119894=

(119899)

sum

(0)isinΓ119899

119886120572120585120572

| 119886120572isin119860

(98)


119894oplusDO

119894


119894 119863119874

119894) formaMa-









119891(A) where Der

119891(A)




(119886 119886) 997891997888rarr (119887 997891997888rarr119891 (120601 (119887) 119886 119886)) (99)



(119887 sdot 120573 ()) = minus120573 (119887) (100)



lowast

(119887 + 119886 +120573) ( + 119886 +120573)

= 119887 + 120601 (119887) 119886 minus 120601 () 119886 + 119886119886 +119908 (119886 119886) + 119887 sdot 120573

minus sdot 120573

(101)


onA119861

119891119861(119887 + 119886 +120573 + 119886 + 120573) = 120573 () + 120573 (119887) +119891 (119886 119886) (102)

Thepair (A119861 119891




119861follows


119908(119886 [1198861015840

119886]) = 119908 (1198861015840

[119886 119886]) minus119908 (119886 [119886 1198861015840

])

119887 sdot 119908 (1198861015840

119886) = 119908 (1198861015840

120601 (119887) sdot 119886) +119908 (120601 (119887) sdot 1198861015840

119886)

(103)





lowast with bracket

[(1198871 1198711 1205731) (1198872 1198712 1205732)] = (0 1198871 [log 120585 1198712]

minus 1198872 [log 120585 1198711] + [1198711 1198712] ⟨[log 120585 1198711] 1198712⟩)(104)


⟨(1198871 1198711 1205731) (1198872 1198712 1205732)⟩

= 12057311198872 +12057321198871 + res (11987111198712) (105)




120590)



120590oplus INT

120590) in which

DO = 120572119888 +

119899

sum

119894=0119886119894120585119894

| 119886119894isinA 120572 isinR


119894=minusinfin

119886119894120585119894

| 119886119894isinA 120573 isinR

(106)



(Ψ119863119874119863119874 119868119873119879)

(Ψ120590119863119874119863119874

120590 119868119873119879

120590)

(107)

are Manin triples




119899DO = Ψ


119894



Ψ119899DO = INT

119894oplus DO

119894 (108)

where

INT119894=

120573 log 120585119894+

(minus1)sum


119886120579120585120579

| 119886120579isin119860

DO119894=

120572119888119894+

(119899)

sum

(0)isinΓ119899

119886120579120585120579

| 119886120579isin119860

(109)



119894 119863119874

119894) form a Ma-






decomposition

Ψ119899DO = Ψ




[(119860 120572 120573) (119861 120572 120573)] = ([119860 119861] + [120573 sdot log 120585 119860]

minus [120573 sdot log 120585 119861] 119888 (119860 119861) 0) (111)

and bilinear form

⟨119860 119861⟩ = res (119860 119861) + 120572 sdot 120573 +120573 sdot 120572 (112)


119899) 0 = (0 0) isin R119899 and 119888(119860 119861) = (1198881(119860 119861)





⟨]120575119865

120575120583⟩ =

119889

119889120598

10038161003816100381610038161003816100381610038161003816120598=0119865 (120583 + 120598]) (113)



119865 119866 (120583) = ⟨120583 [120575119865

120575120583120575119866

120575120583]⟩ (114)



119889120583

119889119905= ⟨120583 [sdot

120575119867

120575120583]⟩ (115)


119877on g by

[119875 119876]119877= [119877 (119875) 119876] + [119875 119877 (119876)] 119876 119875 isin g (116)





+minus



plusmn We write

119875plusmn= 120587


119877= [119875

+ 119876

+]minus[119875

minus 119876

minus] and

119865 119866119877= ⟨120583 [

120575119865

120575120583120575119866

120575120583]

119877

⟩ (117)


⟨120583 [120575119865

120575120583 sdot]⟩ = 0 (118)


119865 119866119877= ⟨120583 [(

120575119865

120575120583)

+

120575119866

120575120583]⟩ (119)


119883119867(120583) sdot 119865 = ⟨120583 [minus(

120575119867

120575120583)

+

120575119865

120575120583]⟩ (120)


⟨119875 [119876 119877]⟩ = ⟨[119877 119875] 119876⟩ (121)


119889119875

119889119905= [(

120575119867

120575120583)

+

119875] (122)



⟨[119875 119876] 119878⟩ = ⟨[119878 119875] 119876⟩ 119878 119875 119876 isin ΨDO (123)


minusoplus ΨDO

+allows us to




Ψ119863119874 Then

120575119867119896

120575119871= 119896119871

119896minus1 (124)


119867119896 119867

119897119877(119871) = 0 for any 119877-matrix 119877




119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (125)


120590 DO

120590 and INT

120590


120590as



120590 The


119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (126)



119899DO DO

119899 and INT

119899allows us


[ ]119877 Once more we set 119867

119896(119871) = res(119871119896) for 119871 isin Ψ

119899DO


120597119871

120597119905= [(119896119871

119896minus1)+

119871] (127)




⟨(119860 120572) (119861 120573)⟩ = ⟨119860 119861⟩ + 120572120573 = res (119860119861) + 120572120573 (128)




⟨120597119871

120597119905 119876⟩ = ⟨[

120597119867

120597119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (129)

119889119897

119889119905= 0 (130)



119889 ( )

119889119905sdot (119876 119903) = ⟨( ) [(119876 119903)

120575119867

120575 ( )]⟩

= ⟨( ) [(119876 119903) (120575119867

120575120575119867

120575)]⟩

= ⟨( ) ([119876120575119867

120575] 119888 (119876

120575119867

120575))⟩

(131)


⟨120597 (119871 119897)

120597119905 (119876 119903)⟩

= ⟨(119871 119897) ([119876120575119867

120575119871] 119888 (119876

120575119867

120575119871))⟩

= ⟨119871 [119876120575119867

120575119871]⟩+ 119897119888 (119876

120575119867

120575119871)

(132)


⟨120597119871

120597119905 119876⟩+ 119903

120597119897

120597119905= ⟨[

120575119867

120575119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (133)





res ([log 120585 119886120585] 119887120585) = res (119887120585 [log 120585 119886120585])

= res(119887120585infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 1205851minus119895)

= res(infin

sum

119895=0

(minus1)119895+1

119895(119887120575

119895

(119886) 1205852minus119895

+ 119887120575119895+1

(119886) 1205851minus119895))

=(minus1)3+1

3119887120575

3(119886) +

(minus1)2+1

2119887120575

3(119886)

= minus16120591 (119887120575

3(119886))

(134)


⟨(119886120585 120572) (119887120585 120573)⟩ = 120591 (119886119887) + 120572120573 (135)


⟨119889119906

119889119905120585 119887120585⟩ = ⟨[ℎ (119906) 120585 119906120585] 119887120585⟩

minus16119897120591 (119887120575

3(ℎ (119906)))

(136)


119889119906

119889119905= ℎ (119906) 120575 (119906) minus 119906120575 (ℎ (119906)) minus (

16) 119897120575

3(ℎ (119906)) (137)




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















algebras 119863119874120590and 119868119873119879



120590and 119868119873119879

120590

vanish








of ΨDO120590by

ΨDO120590119898

=

119872

sum

119894=119898

119886119894120585119894

| 119886119894isinA (61)




Γ119899= 120572 = (1205721 120572119894 120572119899) 120572119894 isinZ

Γ+

119899= 120572 isin Γ

119899 120572

119894isinZ

+

1le 119894 le 119899

120576119894= (0 0 1 0 0)

120599 =

119899

sum

119894=1120576119894isin Γ

119899

120572 =

119899

prod

119894=1120572119894 120572

119894isin Γ

+

119899

(62)


119899 120575120574(119887) = 12057512057411 ∘ sdot sdot sdot ∘ 120575

120574119899


1205751 120575119899 and

(120572

120574) = prod

119894

(120572119894

120574119894

) (63)


119894 with 119894 = 1 119899




120585119894119886 = 119886120585

119894+ 120575

119894(119886) (64)



119899


119886120585120572

sdot 119887120585120573

= sum

120574isinΓ+

119899

(120572

120574)119886120575

120574

(119887) 120585120572+120573minus120574

(65)


119899DOof formal


120585minus1119894119886 = sum

119895ge0(minus1)119895 120575119895

119894(119886) 120585


(66)


commutator

[119860 119861] = 119860119861minus119861119860

= sum

120574isinΓ+

119899

(1120574)(

120597120574

120597120585120574(119860) 120575

120574

(119861) minus120597120574

120597120585120574(119861) 120575

120574

(119860)) (67)


120597120574

120597120585120574(119886120585

120572

) = 120574 (120572

120574) 119886120585

120572minus120574

120575120574

(119886120585120572

) = 120575120574

(119886) 120585120572

(68)





119894 sdot]

considered in (67)

[log 120585119894 119883] = sum

120574isinΓ+

119899

1120574

120597120574

120597120585120574(log 120585

119894) 120575

120574

(119883)

= sum

120574119894ge0

1120574119894

120597120574

119894

120597120585120574119894

119894

(log 120585119894) 120575

120574119894 (119883)

(69)






[log 120585119894 119860119861] = [log 120585

119894 119860] 119861 +119860 [log 120585

119894 119861] (70)






119894(119886)) =


120591 (119886120575120574

(119887)) = (minus1)120574 120591 (119887120575120574 (119886)) (71)




res Ψ119899119863119874 rarr CΨ

119899119863119874 rarr C defined by


119886120572120585120572

) = 120591 (119886minus120599) (72)

is a trace on Ψ119899119863119874


res (119886120585120572119887120585120573) = res (119887120585120573119886120585120572) (73)


119894 120573

119895 we have 120572

119894 120573

119895ge 0

or 120572119894 120573



119894lt 0

If 120572119894+ 120573


Now suppose that 120572119894+ 120573


119894= 120572

119894+ 120573

119894 We have

res (119886120585120572119887120585120573) = (120572

119896 + 120599) 120591 (119886120575

119896+120599

) (74)


res (119887120585120573119886120585120572) = 120591((120573

119896 + 120599) 119887120575

119896+120599

(119886))

= (119896 minus 120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (minus1)119896+120599 (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

= (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

(75)





119894 119860]119861) 119894 =



119894 sdot] is


119894([119862 119860] 119861) +

119888119894([119861 119862] 119860) = 0




119894= 119898 gt 0 for fixed 1 le 119894 le

119899 and

Ψ119899DO

120572119894

=

sum

120573isinΓ119899

119887120573120585120573

= sum

(1205731 120573119899)isinΓ119899

11988612057312058512057311 sdot sdot sdot 120585

120573119899

119899

isinΨ119899DO 120573

119894ge120572

119894

(76)




120572119894


119899119863119874

120572119894




119899DOlowast

120572119894

≃ sum120579isinΓ119899

119897120579120585120579

120579119894le minus120572

119894minus 1 Let 119860 119861 isin

Ψ119899DO

120572119894



120572119894


120572119894

we have

119888 (119860 119861) = res ([119860 119861] 119871)

= res([

[

sum

120572isinΓ119899

119886120572120585120572

sum

120573isinΓ119899

119887120573120585120573]

]

sum

120579isinΓ119899

120579119894leminus120572119894minus1

119897120579120585120579

)


(])(119887120573) 120575

(120583)

(119897120579)

sdot 120585120572+120573+120579minus(]+120583+120593)


(77)


120572119894+120573

119894+ 120579

119894+ (]

119894+120583

119894+120593

119894) = minus 1 (78)


119888 (119860 119861) = res ([log 120585119894 119860] 119861)

= res([log 120585119894 sum

120572isinΓ119899

119886120572120585120572

] sum

120573isinΓ119899

119887120573120585120573

)


(119886120572) 120575

(120591119894)

(119887120573) 120585

120572+120573minus(120595119894+120591)

)


(79)


120572119894+120573

119894+ (120591

119894+120593

119894) = minus 1 (80)


119894= 120579


120579119894le minus120572

119894minus 1 and 120583

119894ge 0


119894


119899DO



120573isinΓ119899

119887120573120585120573 such that 120573 ge 120572






120572




119894= 120579



Let 119875+119899

= sum120572120582120572119909120572

| 120572 isin Γ+




120582120572119909120572 such that the





119894(119909

120572

) =

120572119894119909120572minus120598119894 with 120572 isin Γ

119899





120572isinΓ+

119899

1120572(119906120575

120572

(V) otimes 120575120572 (119891) 119892) (81)



have the following


119899119863119874 and


Ψ119899119863119874 997888rarr Uotimes119875

119899

119906120585120572

997891997888rarr 119906otimes119909120572

(82)


119899119863119874 andU otimes 119875

119899




+119909120573

minus≃ 119909

120572

otimes119909120573


sum

120572120573isinΓ119899

120582120572120573119909120572

+119909120573

minus (83)


[119909120572

+119909120573

minus 119909

120572

+119909120573

minus] = sum

120574isinΓ+

119899

(1120574) (120597

minusand 120597

+) (119909

120572

+119909120573

minus 119909

120572

+119909120573

minus) (84)


119899as

120597120574

minus(119909

120572

+119909120573

minus) = 120574 (

120573

120574)119909

120572

+119909120573minus120574

minus

120597120574

+(119909

120572

+119909120573

minus) = 120574 (

120572

120574)119909

120572minus120574

+119909120573

minus

(85)


+= 120597

minusotimes 120597

+minus 120597

+otimes 120597

minus


1198910(119909minus120599

) = 1 and 1198910(119909120572

) = 0 if 120572 = minus120599



this section



119899 1198671

(H119899H



minus119894 sdot] 119894 = 1 119899


+= 119909

+1 and 119909minus

= 119909minus1 We








119863 = 120582 [log119909+ sdot] + 120573 [log119909


for some 120582 120573


+) = sum119889

+

119899119898119909119899

+119909119898

minusand

119863(119909minus) = sum119889

minus

119899119898119909119899

+119909119898

minus We have

1 = [119909minus 119909

+] 997904rArr 120582 = [119863 (119909

minus) 119909

+] + [119909

minus 119863 (119909

+)]

997904rArr 120582 = sum119889minus

119899119898[119909

119899

+119909119898

minus 119909

+] +sum119889

+

119899119898[119909

minus 119909

119899

+119909119898

minus]

997904rArr 120582 = sum119889minus

119899119898119898119909

119899


+sum119889+

119899119898119899119909

119899minus1+119909119898

minus

(87)



+) = sum119889

+

0119898119909119898


that119863(119909minus) = sum119889

minus

1198990119909119899


119889minus

119899119898119898119909

119899


= minus119889119899119898119899119909

119899minus1+119909119898

minus(88)


(119898+ 1) 119889minus119899119898+1 = minus (119899 + 1) 119889119899+1119898 (89)


Then

119863(119909+) = sum

119898 =0minus1minus 119889

minus

0119898119909119898


minus1minus+119889

+

00 (90)

119863(119909minus) = sum

119899 =0minus1119889minus

1198990119909119899

++119889

minus10119909minus1++119889

minus

00 (91)

Now we write

119863(119909+) = 120582 [log119909

+ 119909

+] + 120573 [log119909

minus 119909

+] minus [120596 119909

+]


+]

(92)


[120596 119909+] = sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120597minus(120596)

= sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120596

= sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

+119889+

00119909minus +

(93)


119863(119909minus) = 120582 [log119909

+ 119909

minus] + 120573 [log119909

minus 119909

minus] minus [120596 119909

minus]

= 120582119909minus1+minus [120596 119909

minus]

(94)



119899 =0minus1119889minus

1198990119909119899

++minus119889

+

00 997904rArr 120597minus(120596)

= minus sum

119899 =0minus1119889minus

1198990119909119899

+minus119889

+

00 997904rArr 120596

= minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

minus119889minus

00119909+ +120596

(95)


120596 = minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

+ sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

minus119889minus

00119909+ minus119889minus

00119909minus

(96)









119894= res([log 120585

plusmn119894 119860]119861) and







4 Applications



+form a Manin







+and g



minus

vanish



120590 DO

120590 and


form ⟨119860 119861⟩120590



that 119886120572119899





INT119899= 119871 isinΨ

119899DO | ord (119871) lt 0

DO119899= 119871 isinΨ

119899DO | ord (119871) ge 0

(97)



119899oplus DO

119899


119899119863119874 119868119873119879

119899 119863119874








119899DO


INT119894=

(minus1)sum


119886120572120585120572

| 119886120572isin119860

DO119894=

(119899)

sum

(0)isinΓ119899

119886120572120585120572

| 119886120572isin119860

(98)


119894oplusDO

119894


119894 119863119874

119894) formaMa-









119891(A) where Der

119891(A)




(119886 119886) 997891997888rarr (119887 997891997888rarr119891 (120601 (119887) 119886 119886)) (99)



(119887 sdot 120573 ()) = minus120573 (119887) (100)



lowast

(119887 + 119886 +120573) ( + 119886 +120573)

= 119887 + 120601 (119887) 119886 minus 120601 () 119886 + 119886119886 +119908 (119886 119886) + 119887 sdot 120573

minus sdot 120573

(101)


onA119861

119891119861(119887 + 119886 +120573 + 119886 + 120573) = 120573 () + 120573 (119887) +119891 (119886 119886) (102)

Thepair (A119861 119891




119861follows


119908(119886 [1198861015840

119886]) = 119908 (1198861015840

[119886 119886]) minus119908 (119886 [119886 1198861015840

])

119887 sdot 119908 (1198861015840

119886) = 119908 (1198861015840

120601 (119887) sdot 119886) +119908 (120601 (119887) sdot 1198861015840

119886)

(103)





lowast with bracket

[(1198871 1198711 1205731) (1198872 1198712 1205732)] = (0 1198871 [log 120585 1198712]

minus 1198872 [log 120585 1198711] + [1198711 1198712] ⟨[log 120585 1198711] 1198712⟩)(104)


⟨(1198871 1198711 1205731) (1198872 1198712 1205732)⟩

= 12057311198872 +12057321198871 + res (11987111198712) (105)




120590)



120590oplus INT

120590) in which

DO = 120572119888 +

119899

sum

119894=0119886119894120585119894

| 119886119894isinA 120572 isinR


119894=minusinfin

119886119894120585119894

| 119886119894isinA 120573 isinR

(106)



(Ψ119863119874119863119874 119868119873119879)

(Ψ120590119863119874119863119874

120590 119868119873119879

120590)

(107)

are Manin triples




119899DO = Ψ


119894



Ψ119899DO = INT

119894oplus DO

119894 (108)

where

INT119894=

120573 log 120585119894+

(minus1)sum


119886120579120585120579

| 119886120579isin119860

DO119894=

120572119888119894+

(119899)

sum

(0)isinΓ119899

119886120579120585120579

| 119886120579isin119860

(109)



119894 119863119874

119894) form a Ma-






decomposition

Ψ119899DO = Ψ




[(119860 120572 120573) (119861 120572 120573)] = ([119860 119861] + [120573 sdot log 120585 119860]

minus [120573 sdot log 120585 119861] 119888 (119860 119861) 0) (111)

and bilinear form

⟨119860 119861⟩ = res (119860 119861) + 120572 sdot 120573 +120573 sdot 120572 (112)


119899) 0 = (0 0) isin R119899 and 119888(119860 119861) = (1198881(119860 119861)





⟨]120575119865

120575120583⟩ =

119889

119889120598

10038161003816100381610038161003816100381610038161003816120598=0119865 (120583 + 120598]) (113)



119865 119866 (120583) = ⟨120583 [120575119865

120575120583120575119866

120575120583]⟩ (114)



119889120583

119889119905= ⟨120583 [sdot

120575119867

120575120583]⟩ (115)


119877on g by

[119875 119876]119877= [119877 (119875) 119876] + [119875 119877 (119876)] 119876 119875 isin g (116)





+minus



plusmn We write

119875plusmn= 120587


119877= [119875

+ 119876

+]minus[119875

minus 119876

minus] and

119865 119866119877= ⟨120583 [

120575119865

120575120583120575119866

120575120583]

119877

⟩ (117)


⟨120583 [120575119865

120575120583 sdot]⟩ = 0 (118)


119865 119866119877= ⟨120583 [(

120575119865

120575120583)

+

120575119866

120575120583]⟩ (119)


119883119867(120583) sdot 119865 = ⟨120583 [minus(

120575119867

120575120583)

+

120575119865

120575120583]⟩ (120)


⟨119875 [119876 119877]⟩ = ⟨[119877 119875] 119876⟩ (121)


119889119875

119889119905= [(

120575119867

120575120583)

+

119875] (122)



⟨[119875 119876] 119878⟩ = ⟨[119878 119875] 119876⟩ 119878 119875 119876 isin ΨDO (123)


minusoplus ΨDO

+allows us to




Ψ119863119874 Then

120575119867119896

120575119871= 119896119871

119896minus1 (124)


119867119896 119867

119897119877(119871) = 0 for any 119877-matrix 119877




119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (125)


120590 DO

120590 and INT

120590


120590as



120590 The


119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (126)



119899DO DO

119899 and INT

119899allows us


[ ]119877 Once more we set 119867

119896(119871) = res(119871119896) for 119871 isin Ψ

119899DO


120597119871

120597119905= [(119896119871

119896minus1)+

119871] (127)




⟨(119860 120572) (119861 120573)⟩ = ⟨119860 119861⟩ + 120572120573 = res (119860119861) + 120572120573 (128)




⟨120597119871

120597119905 119876⟩ = ⟨[

120597119867

120597119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (129)

119889119897

119889119905= 0 (130)



119889 ( )

119889119905sdot (119876 119903) = ⟨( ) [(119876 119903)

120575119867

120575 ( )]⟩

= ⟨( ) [(119876 119903) (120575119867

120575120575119867

120575)]⟩

= ⟨( ) ([119876120575119867

120575] 119888 (119876

120575119867

120575))⟩

(131)


⟨120597 (119871 119897)

120597119905 (119876 119903)⟩

= ⟨(119871 119897) ([119876120575119867

120575119871] 119888 (119876

120575119867

120575119871))⟩

= ⟨119871 [119876120575119867

120575119871]⟩+ 119897119888 (119876

120575119867

120575119871)

(132)


⟨120597119871

120597119905 119876⟩+ 119903

120597119897

120597119905= ⟨[

120575119867

120575119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (133)





res ([log 120585 119886120585] 119887120585) = res (119887120585 [log 120585 119886120585])

= res(119887120585infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 1205851minus119895)

= res(infin

sum

119895=0

(minus1)119895+1

119895(119887120575

119895

(119886) 1205852minus119895

+ 119887120575119895+1

(119886) 1205851minus119895))

=(minus1)3+1

3119887120575

3(119886) +

(minus1)2+1

2119887120575

3(119886)

= minus16120591 (119887120575

3(119886))

(134)


⟨(119886120585 120572) (119887120585 120573)⟩ = 120591 (119886119887) + 120572120573 (135)


⟨119889119906

119889119905120585 119887120585⟩ = ⟨[ℎ (119906) 120585 119906120585] 119887120585⟩

minus16119897120591 (119887120575

3(ℎ (119906)))

(136)


119889119906

119889119905= ℎ (119906) 120575 (119906) minus 119906120575 (ℎ (119906)) minus (

16) 119897120575

3(ℎ (119906)) (137)




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












res Ψ119899119863119874 rarr CΨ

119899119863119874 rarr C defined by


119886120572120585120572

) = 120591 (119886minus120599) (72)

is a trace on Ψ119899119863119874


res (119886120585120572119887120585120573) = res (119887120585120573119886120585120572) (73)


119894 120573

119895 we have 120572

119894 120573

119895ge 0

or 120572119894 120573



119894lt 0

If 120572119894+ 120573


Now suppose that 120572119894+ 120573


119894= 120572

119894+ 120573

119894 We have

res (119886120585120572119887120585120573) = (120572

119896 + 120599) 120591 (119886120575

119896+120599

) (74)


res (119887120585120573119886120585120572) = 120591((120573

119896 + 120599) 119887120575

119896+120599

(119886))

= (119896 minus 120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (120572

119896 + 120599) 120591 (119887120575

119896+120599

(119886))

= (minus1)119896+120599 (minus1)119896+120599 (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

= (120572

119896 + 120599) 120591 (119886120575

119896+120599

(119887))

(75)





119894 119860]119861) 119894 =



119894 sdot] is


119894([119862 119860] 119861) +

119888119894([119861 119862] 119860) = 0




119894= 119898 gt 0 for fixed 1 le 119894 le

119899 and

Ψ119899DO

120572119894

=

sum

120573isinΓ119899

119887120573120585120573

= sum

(1205731 120573119899)isinΓ119899

11988612057312058512057311 sdot sdot sdot 120585

120573119899

119899

isinΨ119899DO 120573

119894ge120572

119894

(76)




120572119894


119899119863119874

120572119894




119899DOlowast

120572119894

≃ sum120579isinΓ119899

119897120579120585120579

120579119894le minus120572

119894minus 1 Let 119860 119861 isin

Ψ119899DO

120572119894



120572119894


120572119894

we have

119888 (119860 119861) = res ([119860 119861] 119871)

= res([

[

sum

120572isinΓ119899

119886120572120585120572

sum

120573isinΓ119899

119887120573120585120573]

]

sum

120579isinΓ119899

120579119894leminus120572119894minus1

119897120579120585120579

)


(])(119887120573) 120575

(120583)

(119897120579)

sdot 120585120572+120573+120579minus(]+120583+120593)


(77)


120572119894+120573

119894+ 120579

119894+ (]

119894+120583

119894+120593

119894) = minus 1 (78)


119888 (119860 119861) = res ([log 120585119894 119860] 119861)

= res([log 120585119894 sum

120572isinΓ119899

119886120572120585120572

] sum

120573isinΓ119899

119887120573120585120573

)


(119886120572) 120575

(120591119894)

(119887120573) 120585

120572+120573minus(120595119894+120591)

)


(79)


120572119894+120573

119894+ (120591

119894+120593

119894) = minus 1 (80)


119894= 120579


120579119894le minus120572

119894minus 1 and 120583

119894ge 0


119894


119899DO



120573isinΓ119899

119887120573120585120573 such that 120573 ge 120572






120572




119894= 120579



Let 119875+119899

= sum120572120582120572119909120572

| 120572 isin Γ+




120582120572119909120572 such that the





119894(119909

120572

) =

120572119894119909120572minus120598119894 with 120572 isin Γ

119899





120572isinΓ+

119899

1120572(119906120575

120572

(V) otimes 120575120572 (119891) 119892) (81)



have the following


119899119863119874 and


Ψ119899119863119874 997888rarr Uotimes119875

119899

119906120585120572

997891997888rarr 119906otimes119909120572

(82)


119899119863119874 andU otimes 119875

119899




+119909120573

minus≃ 119909

120572

otimes119909120573


sum

120572120573isinΓ119899

120582120572120573119909120572

+119909120573

minus (83)


[119909120572

+119909120573

minus 119909

120572

+119909120573

minus] = sum

120574isinΓ+

119899

(1120574) (120597

minusand 120597

+) (119909

120572

+119909120573

minus 119909

120572

+119909120573

minus) (84)


119899as

120597120574

minus(119909

120572

+119909120573

minus) = 120574 (

120573

120574)119909

120572

+119909120573minus120574

minus

120597120574

+(119909

120572

+119909120573

minus) = 120574 (

120572

120574)119909

120572minus120574

+119909120573

minus

(85)


+= 120597

minusotimes 120597

+minus 120597

+otimes 120597

minus


1198910(119909minus120599

) = 1 and 1198910(119909120572

) = 0 if 120572 = minus120599



this section



119899 1198671

(H119899H



minus119894 sdot] 119894 = 1 119899


+= 119909

+1 and 119909minus

= 119909minus1 We








119863 = 120582 [log119909+ sdot] + 120573 [log119909


for some 120582 120573


+) = sum119889

+

119899119898119909119899

+119909119898

minusand

119863(119909minus) = sum119889

minus

119899119898119909119899

+119909119898

minus We have

1 = [119909minus 119909

+] 997904rArr 120582 = [119863 (119909

minus) 119909

+] + [119909

minus 119863 (119909

+)]

997904rArr 120582 = sum119889minus

119899119898[119909

119899

+119909119898

minus 119909

+] +sum119889

+

119899119898[119909

minus 119909

119899

+119909119898

minus]

997904rArr 120582 = sum119889minus

119899119898119898119909

119899


+sum119889+

119899119898119899119909

119899minus1+119909119898

minus

(87)



+) = sum119889

+

0119898119909119898


that119863(119909minus) = sum119889

minus

1198990119909119899


119889minus

119899119898119898119909

119899


= minus119889119899119898119899119909

119899minus1+119909119898

minus(88)


(119898+ 1) 119889minus119899119898+1 = minus (119899 + 1) 119889119899+1119898 (89)


Then

119863(119909+) = sum

119898 =0minus1minus 119889

minus

0119898119909119898


minus1minus+119889

+

00 (90)

119863(119909minus) = sum

119899 =0minus1119889minus

1198990119909119899

++119889

minus10119909minus1++119889

minus

00 (91)

Now we write

119863(119909+) = 120582 [log119909

+ 119909

+] + 120573 [log119909

minus 119909

+] minus [120596 119909

+]


+]

(92)


[120596 119909+] = sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120597minus(120596)

= sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120596

= sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

+119889+

00119909minus +

(93)


119863(119909minus) = 120582 [log119909

+ 119909

minus] + 120573 [log119909

minus 119909

minus] minus [120596 119909

minus]

= 120582119909minus1+minus [120596 119909

minus]

(94)



119899 =0minus1119889minus

1198990119909119899

++minus119889

+

00 997904rArr 120597minus(120596)

= minus sum

119899 =0minus1119889minus

1198990119909119899

+minus119889

+

00 997904rArr 120596

= minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

minus119889minus

00119909+ +120596

(95)


120596 = minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

+ sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

minus119889minus

00119909+ minus119889minus

00119909minus

(96)









119894= res([log 120585

plusmn119894 119860]119861) and







4 Applications



+form a Manin







+and g



minus

vanish



120590 DO

120590 and


form ⟨119860 119861⟩120590



that 119886120572119899





INT119899= 119871 isinΨ

119899DO | ord (119871) lt 0

DO119899= 119871 isinΨ

119899DO | ord (119871) ge 0

(97)



119899oplus DO

119899


119899119863119874 119868119873119879

119899 119863119874








119899DO


INT119894=

(minus1)sum


119886120572120585120572

| 119886120572isin119860

DO119894=

(119899)

sum

(0)isinΓ119899

119886120572120585120572

| 119886120572isin119860

(98)


119894oplusDO

119894


119894 119863119874

119894) formaMa-









119891(A) where Der

119891(A)




(119886 119886) 997891997888rarr (119887 997891997888rarr119891 (120601 (119887) 119886 119886)) (99)



(119887 sdot 120573 ()) = minus120573 (119887) (100)



lowast

(119887 + 119886 +120573) ( + 119886 +120573)

= 119887 + 120601 (119887) 119886 minus 120601 () 119886 + 119886119886 +119908 (119886 119886) + 119887 sdot 120573

minus sdot 120573

(101)


onA119861

119891119861(119887 + 119886 +120573 + 119886 + 120573) = 120573 () + 120573 (119887) +119891 (119886 119886) (102)

Thepair (A119861 119891




119861follows


119908(119886 [1198861015840

119886]) = 119908 (1198861015840

[119886 119886]) minus119908 (119886 [119886 1198861015840

])

119887 sdot 119908 (1198861015840

119886) = 119908 (1198861015840

120601 (119887) sdot 119886) +119908 (120601 (119887) sdot 1198861015840

119886)

(103)





lowast with bracket

[(1198871 1198711 1205731) (1198872 1198712 1205732)] = (0 1198871 [log 120585 1198712]

minus 1198872 [log 120585 1198711] + [1198711 1198712] ⟨[log 120585 1198711] 1198712⟩)(104)


⟨(1198871 1198711 1205731) (1198872 1198712 1205732)⟩

= 12057311198872 +12057321198871 + res (11987111198712) (105)




120590)



120590oplus INT

120590) in which

DO = 120572119888 +

119899

sum

119894=0119886119894120585119894

| 119886119894isinA 120572 isinR


119894=minusinfin

119886119894120585119894

| 119886119894isinA 120573 isinR

(106)



(Ψ119863119874119863119874 119868119873119879)

(Ψ120590119863119874119863119874

120590 119868119873119879

120590)

(107)

are Manin triples




119899DO = Ψ


119894



Ψ119899DO = INT

119894oplus DO

119894 (108)

where

INT119894=

120573 log 120585119894+

(minus1)sum


119886120579120585120579

| 119886120579isin119860

DO119894=

120572119888119894+

(119899)

sum

(0)isinΓ119899

119886120579120585120579

| 119886120579isin119860

(109)



119894 119863119874

119894) form a Ma-






decomposition

Ψ119899DO = Ψ




[(119860 120572 120573) (119861 120572 120573)] = ([119860 119861] + [120573 sdot log 120585 119860]

minus [120573 sdot log 120585 119861] 119888 (119860 119861) 0) (111)

and bilinear form

⟨119860 119861⟩ = res (119860 119861) + 120572 sdot 120573 +120573 sdot 120572 (112)


119899) 0 = (0 0) isin R119899 and 119888(119860 119861) = (1198881(119860 119861)





⟨]120575119865

120575120583⟩ =

119889

119889120598

10038161003816100381610038161003816100381610038161003816120598=0119865 (120583 + 120598]) (113)



119865 119866 (120583) = ⟨120583 [120575119865

120575120583120575119866

120575120583]⟩ (114)



119889120583

119889119905= ⟨120583 [sdot

120575119867

120575120583]⟩ (115)


119877on g by

[119875 119876]119877= [119877 (119875) 119876] + [119875 119877 (119876)] 119876 119875 isin g (116)





+minus



plusmn We write

119875plusmn= 120587


119877= [119875

+ 119876

+]minus[119875

minus 119876

minus] and

119865 119866119877= ⟨120583 [

120575119865

120575120583120575119866

120575120583]

119877

⟩ (117)


⟨120583 [120575119865

120575120583 sdot]⟩ = 0 (118)


119865 119866119877= ⟨120583 [(

120575119865

120575120583)

+

120575119866

120575120583]⟩ (119)


119883119867(120583) sdot 119865 = ⟨120583 [minus(

120575119867

120575120583)

+

120575119865

120575120583]⟩ (120)


⟨119875 [119876 119877]⟩ = ⟨[119877 119875] 119876⟩ (121)


119889119875

119889119905= [(

120575119867

120575120583)

+

119875] (122)



⟨[119875 119876] 119878⟩ = ⟨[119878 119875] 119876⟩ 119878 119875 119876 isin ΨDO (123)


minusoplus ΨDO

+allows us to




Ψ119863119874 Then

120575119867119896

120575119871= 119896119871

119896minus1 (124)


119867119896 119867

119897119877(119871) = 0 for any 119877-matrix 119877




119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (125)


120590 DO

120590 and INT

120590


120590as



120590 The


119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (126)



119899DO DO

119899 and INT

119899allows us


[ ]119877 Once more we set 119867

119896(119871) = res(119871119896) for 119871 isin Ψ

119899DO


120597119871

120597119905= [(119896119871

119896minus1)+

119871] (127)




⟨(119860 120572) (119861 120573)⟩ = ⟨119860 119861⟩ + 120572120573 = res (119860119861) + 120572120573 (128)




⟨120597119871

120597119905 119876⟩ = ⟨[

120597119867

120597119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (129)

119889119897

119889119905= 0 (130)



119889 ( )

119889119905sdot (119876 119903) = ⟨( ) [(119876 119903)

120575119867

120575 ( )]⟩

= ⟨( ) [(119876 119903) (120575119867

120575120575119867

120575)]⟩

= ⟨( ) ([119876120575119867

120575] 119888 (119876

120575119867

120575))⟩

(131)


⟨120597 (119871 119897)

120597119905 (119876 119903)⟩

= ⟨(119871 119897) ([119876120575119867

120575119871] 119888 (119876

120575119867

120575119871))⟩

= ⟨119871 [119876120575119867

120575119871]⟩+ 119897119888 (119876

120575119867

120575119871)

(132)


⟨120597119871

120597119905 119876⟩+ 119903

120597119897

120597119905= ⟨[

120575119867

120575119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (133)





res ([log 120585 119886120585] 119887120585) = res (119887120585 [log 120585 119886120585])

= res(119887120585infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 1205851minus119895)

= res(infin

sum

119895=0

(minus1)119895+1

119895(119887120575

119895

(119886) 1205852minus119895

+ 119887120575119895+1

(119886) 1205851minus119895))

=(minus1)3+1

3119887120575

3(119886) +

(minus1)2+1

2119887120575

3(119886)

= minus16120591 (119887120575

3(119886))

(134)


⟨(119886120585 120572) (119887120585 120573)⟩ = 120591 (119886119887) + 120572120573 (135)


⟨119889119906

119889119905120585 119887120585⟩ = ⟨[ℎ (119906) 120585 119906120585] 119887120585⟩

minus16119897120591 (119887120575

3(ℎ (119906)))

(136)


119889119906

119889119905= ℎ (119906) 120575 (119906) minus 119906120575 (ℎ (119906)) minus (

16) 119897120575

3(ℎ (119906)) (137)




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














120572




119894= 120579



Let 119875+119899

= sum120572120582120572119909120572

| 120572 isin Γ+




120582120572119909120572 such that the





119894(119909

120572

) =

120572119894119909120572minus120598119894 with 120572 isin Γ

119899





120572isinΓ+

119899

1120572(119906120575

120572

(V) otimes 120575120572 (119891) 119892) (81)



have the following


119899119863119874 and


Ψ119899119863119874 997888rarr Uotimes119875

119899

119906120585120572

997891997888rarr 119906otimes119909120572

(82)


119899119863119874 andU otimes 119875

119899




+119909120573

minus≃ 119909

120572

otimes119909120573


sum

120572120573isinΓ119899

120582120572120573119909120572

+119909120573

minus (83)


[119909120572

+119909120573

minus 119909

120572

+119909120573

minus] = sum

120574isinΓ+

119899

(1120574) (120597

minusand 120597

+) (119909

120572

+119909120573

minus 119909

120572

+119909120573

minus) (84)


119899as

120597120574

minus(119909

120572

+119909120573

minus) = 120574 (

120573

120574)119909

120572

+119909120573minus120574

minus

120597120574

+(119909

120572

+119909120573

minus) = 120574 (

120572

120574)119909

120572minus120574

+119909120573

minus

(85)


+= 120597

minusotimes 120597

+minus 120597

+otimes 120597

minus


1198910(119909minus120599

) = 1 and 1198910(119909120572

) = 0 if 120572 = minus120599



this section



119899 1198671

(H119899H



minus119894 sdot] 119894 = 1 119899


+= 119909

+1 and 119909minus

= 119909minus1 We








119863 = 120582 [log119909+ sdot] + 120573 [log119909


for some 120582 120573


+) = sum119889

+

119899119898119909119899

+119909119898

minusand

119863(119909minus) = sum119889

minus

119899119898119909119899

+119909119898

minus We have

1 = [119909minus 119909

+] 997904rArr 120582 = [119863 (119909

minus) 119909

+] + [119909

minus 119863 (119909

+)]

997904rArr 120582 = sum119889minus

119899119898[119909

119899

+119909119898

minus 119909

+] +sum119889

+

119899119898[119909

minus 119909

119899

+119909119898

minus]

997904rArr 120582 = sum119889minus

119899119898119898119909

119899


+sum119889+

119899119898119899119909

119899minus1+119909119898

minus

(87)



+) = sum119889

+

0119898119909119898


that119863(119909minus) = sum119889

minus

1198990119909119899


119889minus

119899119898119898119909

119899


= minus119889119899119898119899119909

119899minus1+119909119898

minus(88)


(119898+ 1) 119889minus119899119898+1 = minus (119899 + 1) 119889119899+1119898 (89)


Then

119863(119909+) = sum

119898 =0minus1minus 119889

minus

0119898119909119898


minus1minus+119889

+

00 (90)

119863(119909minus) = sum

119899 =0minus1119889minus

1198990119909119899

++119889

minus10119909minus1++119889

minus

00 (91)

Now we write

119863(119909+) = 120582 [log119909

+ 119909

+] + 120573 [log119909

minus 119909

+] minus [120596 119909

+]


+]

(92)


[120596 119909+] = sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120597minus(120596)

= sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120596

= sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

+119889+

00119909minus +

(93)


119863(119909minus) = 120582 [log119909

+ 119909

minus] + 120573 [log119909

minus 119909

minus] minus [120596 119909

minus]

= 120582119909minus1+minus [120596 119909

minus]

(94)



119899 =0minus1119889minus

1198990119909119899

++minus119889

+

00 997904rArr 120597minus(120596)

= minus sum

119899 =0minus1119889minus

1198990119909119899

+minus119889

+

00 997904rArr 120596

= minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

minus119889minus

00119909+ +120596

(95)


120596 = minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

+ sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

minus119889minus

00119909+ minus119889minus

00119909minus

(96)









119894= res([log 120585

plusmn119894 119860]119861) and







4 Applications



+form a Manin







+and g



minus

vanish



120590 DO

120590 and


form ⟨119860 119861⟩120590



that 119886120572119899





INT119899= 119871 isinΨ

119899DO | ord (119871) lt 0

DO119899= 119871 isinΨ

119899DO | ord (119871) ge 0

(97)



119899oplus DO

119899


119899119863119874 119868119873119879

119899 119863119874








119899DO


INT119894=

(minus1)sum


119886120572120585120572

| 119886120572isin119860

DO119894=

(119899)

sum

(0)isinΓ119899

119886120572120585120572

| 119886120572isin119860

(98)


119894oplusDO

119894


119894 119863119874

119894) formaMa-









119891(A) where Der

119891(A)




(119886 119886) 997891997888rarr (119887 997891997888rarr119891 (120601 (119887) 119886 119886)) (99)



(119887 sdot 120573 ()) = minus120573 (119887) (100)



lowast

(119887 + 119886 +120573) ( + 119886 +120573)

= 119887 + 120601 (119887) 119886 minus 120601 () 119886 + 119886119886 +119908 (119886 119886) + 119887 sdot 120573

minus sdot 120573

(101)


onA119861

119891119861(119887 + 119886 +120573 + 119886 + 120573) = 120573 () + 120573 (119887) +119891 (119886 119886) (102)

Thepair (A119861 119891




119861follows


119908(119886 [1198861015840

119886]) = 119908 (1198861015840

[119886 119886]) minus119908 (119886 [119886 1198861015840

])

119887 sdot 119908 (1198861015840

119886) = 119908 (1198861015840

120601 (119887) sdot 119886) +119908 (120601 (119887) sdot 1198861015840

119886)

(103)





lowast with bracket

[(1198871 1198711 1205731) (1198872 1198712 1205732)] = (0 1198871 [log 120585 1198712]

minus 1198872 [log 120585 1198711] + [1198711 1198712] ⟨[log 120585 1198711] 1198712⟩)(104)


⟨(1198871 1198711 1205731) (1198872 1198712 1205732)⟩

= 12057311198872 +12057321198871 + res (11987111198712) (105)




120590)



120590oplus INT

120590) in which

DO = 120572119888 +

119899

sum

119894=0119886119894120585119894

| 119886119894isinA 120572 isinR


119894=minusinfin

119886119894120585119894

| 119886119894isinA 120573 isinR

(106)



(Ψ119863119874119863119874 119868119873119879)

(Ψ120590119863119874119863119874

120590 119868119873119879

120590)

(107)

are Manin triples




119899DO = Ψ


119894



Ψ119899DO = INT

119894oplus DO

119894 (108)

where

INT119894=

120573 log 120585119894+

(minus1)sum


119886120579120585120579

| 119886120579isin119860

DO119894=

120572119888119894+

(119899)

sum

(0)isinΓ119899

119886120579120585120579

| 119886120579isin119860

(109)



119894 119863119874

119894) form a Ma-






decomposition

Ψ119899DO = Ψ




[(119860 120572 120573) (119861 120572 120573)] = ([119860 119861] + [120573 sdot log 120585 119860]

minus [120573 sdot log 120585 119861] 119888 (119860 119861) 0) (111)

and bilinear form

⟨119860 119861⟩ = res (119860 119861) + 120572 sdot 120573 +120573 sdot 120572 (112)


119899) 0 = (0 0) isin R119899 and 119888(119860 119861) = (1198881(119860 119861)





⟨]120575119865

120575120583⟩ =

119889

119889120598

10038161003816100381610038161003816100381610038161003816120598=0119865 (120583 + 120598]) (113)



119865 119866 (120583) = ⟨120583 [120575119865

120575120583120575119866

120575120583]⟩ (114)



119889120583

119889119905= ⟨120583 [sdot

120575119867

120575120583]⟩ (115)


119877on g by

[119875 119876]119877= [119877 (119875) 119876] + [119875 119877 (119876)] 119876 119875 isin g (116)





+minus



plusmn We write

119875plusmn= 120587


119877= [119875

+ 119876

+]minus[119875

minus 119876

minus] and

119865 119866119877= ⟨120583 [

120575119865

120575120583120575119866

120575120583]

119877

⟩ (117)


⟨120583 [120575119865

120575120583 sdot]⟩ = 0 (118)


119865 119866119877= ⟨120583 [(

120575119865

120575120583)

+

120575119866

120575120583]⟩ (119)


119883119867(120583) sdot 119865 = ⟨120583 [minus(

120575119867

120575120583)

+

120575119865

120575120583]⟩ (120)


⟨119875 [119876 119877]⟩ = ⟨[119877 119875] 119876⟩ (121)


119889119875

119889119905= [(

120575119867

120575120583)

+

119875] (122)



⟨[119875 119876] 119878⟩ = ⟨[119878 119875] 119876⟩ 119878 119875 119876 isin ΨDO (123)


minusoplus ΨDO

+allows us to




Ψ119863119874 Then

120575119867119896

120575119871= 119896119871

119896minus1 (124)


119867119896 119867

119897119877(119871) = 0 for any 119877-matrix 119877




119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (125)


120590 DO

120590 and INT

120590


120590as



120590 The


119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (126)



119899DO DO

119899 and INT

119899allows us


[ ]119877 Once more we set 119867

119896(119871) = res(119871119896) for 119871 isin Ψ

119899DO


120597119871

120597119905= [(119896119871

119896minus1)+

119871] (127)




⟨(119860 120572) (119861 120573)⟩ = ⟨119860 119861⟩ + 120572120573 = res (119860119861) + 120572120573 (128)




⟨120597119871

120597119905 119876⟩ = ⟨[

120597119867

120597119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (129)

119889119897

119889119905= 0 (130)



119889 ( )

119889119905sdot (119876 119903) = ⟨( ) [(119876 119903)

120575119867

120575 ( )]⟩

= ⟨( ) [(119876 119903) (120575119867

120575120575119867

120575)]⟩

= ⟨( ) ([119876120575119867

120575] 119888 (119876

120575119867

120575))⟩

(131)


⟨120597 (119871 119897)

120597119905 (119876 119903)⟩

= ⟨(119871 119897) ([119876120575119867

120575119871] 119888 (119876

120575119867

120575119871))⟩

= ⟨119871 [119876120575119867

120575119871]⟩+ 119897119888 (119876

120575119867

120575119871)

(132)


⟨120597119871

120597119905 119876⟩+ 119903

120597119897

120597119905= ⟨[

120575119867

120575119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (133)





res ([log 120585 119886120585] 119887120585) = res (119887120585 [log 120585 119886120585])

= res(119887120585infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 1205851minus119895)

= res(infin

sum

119895=0

(minus1)119895+1

119895(119887120575

119895

(119886) 1205852minus119895

+ 119887120575119895+1

(119886) 1205851minus119895))

=(minus1)3+1

3119887120575

3(119886) +

(minus1)2+1

2119887120575

3(119886)

= minus16120591 (119887120575

3(119886))

(134)


⟨(119886120585 120572) (119887120585 120573)⟩ = 120591 (119886119887) + 120572120573 (135)


⟨119889119906

119889119905120585 119887120585⟩ = ⟨[ℎ (119906) 120585 119906120585] 119887120585⟩

minus16119897120591 (119887120575

3(ℎ (119906)))

(136)


119889119906

119889119905= ℎ (119906) 120575 (119906) minus 119906120575 (ℎ (119906)) minus (

16) 119897120575

3(ℎ (119906)) (137)




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Then

119863(119909+) = sum

119898 =0minus1minus 119889

minus

0119898119909119898


minus1minus+119889

+

00 (90)

119863(119909minus) = sum

119899 =0minus1119889minus

1198990119909119899

++119889

minus10119909minus1++119889

minus

00 (91)

Now we write

119863(119909+) = 120582 [log119909

+ 119909

+] + 120573 [log119909

minus 119909

+] minus [120596 119909

+]


+]

(92)


[120596 119909+] = sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120597minus(120596)

= sum

119898 =0minus1119889minus

0119898119909119898

minus+119889

+

00 997904rArr 120596

= sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

+119889+

00119909minus +

(93)


119863(119909minus) = 120582 [log119909

+ 119909

minus] + 120573 [log119909

minus 119909

minus] minus [120596 119909

minus]

= 120582119909minus1+minus [120596 119909

minus]

(94)



119899 =0minus1119889minus

1198990119909119899

++minus119889

+

00 997904rArr 120597minus(120596)

= minus sum

119899 =0minus1119889minus

1198990119909119899

+minus119889

+

00 997904rArr 120596

= minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

minus119889minus

00119909+ +120596

(95)


120596 = minus sum

119899 =0minus1

119889minus

1198990

119899 + 1119909119899+1+

+ sum

119898 =0minus1

119889minus

0119898

119898 + 1119909119898+1minus

minus119889minus

00119909+ minus119889minus

00119909minus

(96)









119894= res([log 120585

plusmn119894 119860]119861) and







4 Applications



+form a Manin







+and g



minus

vanish



120590 DO

120590 and


form ⟨119860 119861⟩120590



that 119886120572119899





INT119899= 119871 isinΨ

119899DO | ord (119871) lt 0

DO119899= 119871 isinΨ

119899DO | ord (119871) ge 0

(97)



119899oplus DO

119899


119899119863119874 119868119873119879

119899 119863119874








119899DO


INT119894=

(minus1)sum


119886120572120585120572

| 119886120572isin119860

DO119894=

(119899)

sum

(0)isinΓ119899

119886120572120585120572

| 119886120572isin119860

(98)


119894oplusDO

119894


119894 119863119874

119894) formaMa-









119891(A) where Der

119891(A)




(119886 119886) 997891997888rarr (119887 997891997888rarr119891 (120601 (119887) 119886 119886)) (99)



(119887 sdot 120573 ()) = minus120573 (119887) (100)



lowast

(119887 + 119886 +120573) ( + 119886 +120573)

= 119887 + 120601 (119887) 119886 minus 120601 () 119886 + 119886119886 +119908 (119886 119886) + 119887 sdot 120573

minus sdot 120573

(101)


onA119861

119891119861(119887 + 119886 +120573 + 119886 + 120573) = 120573 () + 120573 (119887) +119891 (119886 119886) (102)

Thepair (A119861 119891




119861follows


119908(119886 [1198861015840

119886]) = 119908 (1198861015840

[119886 119886]) minus119908 (119886 [119886 1198861015840

])

119887 sdot 119908 (1198861015840

119886) = 119908 (1198861015840

120601 (119887) sdot 119886) +119908 (120601 (119887) sdot 1198861015840

119886)

(103)





lowast with bracket

[(1198871 1198711 1205731) (1198872 1198712 1205732)] = (0 1198871 [log 120585 1198712]

minus 1198872 [log 120585 1198711] + [1198711 1198712] ⟨[log 120585 1198711] 1198712⟩)(104)


⟨(1198871 1198711 1205731) (1198872 1198712 1205732)⟩

= 12057311198872 +12057321198871 + res (11987111198712) (105)




120590)



120590oplus INT

120590) in which

DO = 120572119888 +

119899

sum

119894=0119886119894120585119894

| 119886119894isinA 120572 isinR


119894=minusinfin

119886119894120585119894

| 119886119894isinA 120573 isinR

(106)



(Ψ119863119874119863119874 119868119873119879)

(Ψ120590119863119874119863119874

120590 119868119873119879

120590)

(107)

are Manin triples




119899DO = Ψ


119894



Ψ119899DO = INT

119894oplus DO

119894 (108)

where

INT119894=

120573 log 120585119894+

(minus1)sum


119886120579120585120579

| 119886120579isin119860

DO119894=

120572119888119894+

(119899)

sum

(0)isinΓ119899

119886120579120585120579

| 119886120579isin119860

(109)



119894 119863119874

119894) form a Ma-






decomposition

Ψ119899DO = Ψ




[(119860 120572 120573) (119861 120572 120573)] = ([119860 119861] + [120573 sdot log 120585 119860]

minus [120573 sdot log 120585 119861] 119888 (119860 119861) 0) (111)

and bilinear form

⟨119860 119861⟩ = res (119860 119861) + 120572 sdot 120573 +120573 sdot 120572 (112)


119899) 0 = (0 0) isin R119899 and 119888(119860 119861) = (1198881(119860 119861)





⟨]120575119865

120575120583⟩ =

119889

119889120598

10038161003816100381610038161003816100381610038161003816120598=0119865 (120583 + 120598]) (113)



119865 119866 (120583) = ⟨120583 [120575119865

120575120583120575119866

120575120583]⟩ (114)



119889120583

119889119905= ⟨120583 [sdot

120575119867

120575120583]⟩ (115)


119877on g by

[119875 119876]119877= [119877 (119875) 119876] + [119875 119877 (119876)] 119876 119875 isin g (116)





+minus



plusmn We write

119875plusmn= 120587


119877= [119875

+ 119876

+]minus[119875

minus 119876

minus] and

119865 119866119877= ⟨120583 [

120575119865

120575120583120575119866

120575120583]

119877

⟩ (117)


⟨120583 [120575119865

120575120583 sdot]⟩ = 0 (118)


119865 119866119877= ⟨120583 [(

120575119865

120575120583)

+

120575119866

120575120583]⟩ (119)


119883119867(120583) sdot 119865 = ⟨120583 [minus(

120575119867

120575120583)

+

120575119865

120575120583]⟩ (120)


⟨119875 [119876 119877]⟩ = ⟨[119877 119875] 119876⟩ (121)


119889119875

119889119905= [(

120575119867

120575120583)

+

119875] (122)



⟨[119875 119876] 119878⟩ = ⟨[119878 119875] 119876⟩ 119878 119875 119876 isin ΨDO (123)


minusoplus ΨDO

+allows us to




Ψ119863119874 Then

120575119867119896

120575119871= 119896119871

119896minus1 (124)


119867119896 119867

119897119877(119871) = 0 for any 119877-matrix 119877




119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (125)


120590 DO

120590 and INT

120590


120590as



120590 The


119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (126)



119899DO DO

119899 and INT

119899allows us


[ ]119877 Once more we set 119867

119896(119871) = res(119871119896) for 119871 isin Ψ

119899DO


120597119871

120597119905= [(119896119871

119896minus1)+

119871] (127)




⟨(119860 120572) (119861 120573)⟩ = ⟨119860 119861⟩ + 120572120573 = res (119860119861) + 120572120573 (128)




⟨120597119871

120597119905 119876⟩ = ⟨[

120597119867

120597119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (129)

119889119897

119889119905= 0 (130)



119889 ( )

119889119905sdot (119876 119903) = ⟨( ) [(119876 119903)

120575119867

120575 ( )]⟩

= ⟨( ) [(119876 119903) (120575119867

120575120575119867

120575)]⟩

= ⟨( ) ([119876120575119867

120575] 119888 (119876

120575119867

120575))⟩

(131)


⟨120597 (119871 119897)

120597119905 (119876 119903)⟩

= ⟨(119871 119897) ([119876120575119867

120575119871] 119888 (119876

120575119867

120575119871))⟩

= ⟨119871 [119876120575119867

120575119871]⟩+ 119897119888 (119876

120575119867

120575119871)

(132)


⟨120597119871

120597119905 119876⟩+ 119903

120597119897

120597119905= ⟨[

120575119867

120575119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (133)





res ([log 120585 119886120585] 119887120585) = res (119887120585 [log 120585 119886120585])

= res(119887120585infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 1205851minus119895)

= res(infin

sum

119895=0

(minus1)119895+1

119895(119887120575

119895

(119886) 1205852minus119895

+ 119887120575119895+1

(119886) 1205851minus119895))

=(minus1)3+1

3119887120575

3(119886) +

(minus1)2+1

2119887120575

3(119886)

= minus16120591 (119887120575

3(119886))

(134)


⟨(119886120585 120572) (119887120585 120573)⟩ = 120591 (119886119887) + 120572120573 (135)


⟨119889119906

119889119905120585 119887120585⟩ = ⟨[ℎ (119906) 120585 119906120585] 119887120585⟩

minus16119897120591 (119887120575

3(ℎ (119906)))

(136)


119889119906

119889119905= ℎ (119906) 120575 (119906) minus 119906120575 (ℎ (119906)) minus (

16) 119897120575

3(ℎ (119906)) (137)




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119899DO


INT119894=

(minus1)sum


119886120572120585120572

| 119886120572isin119860

DO119894=

(119899)

sum

(0)isinΓ119899

119886120572120585120572

| 119886120572isin119860

(98)


119894oplusDO

119894


119894 119863119874

119894) formaMa-









119891(A) where Der

119891(A)




(119886 119886) 997891997888rarr (119887 997891997888rarr119891 (120601 (119887) 119886 119886)) (99)



(119887 sdot 120573 ()) = minus120573 (119887) (100)



lowast

(119887 + 119886 +120573) ( + 119886 +120573)

= 119887 + 120601 (119887) 119886 minus 120601 () 119886 + 119886119886 +119908 (119886 119886) + 119887 sdot 120573

minus sdot 120573

(101)


onA119861

119891119861(119887 + 119886 +120573 + 119886 + 120573) = 120573 () + 120573 (119887) +119891 (119886 119886) (102)

Thepair (A119861 119891




119861follows


119908(119886 [1198861015840

119886]) = 119908 (1198861015840

[119886 119886]) minus119908 (119886 [119886 1198861015840

])

119887 sdot 119908 (1198861015840

119886) = 119908 (1198861015840

120601 (119887) sdot 119886) +119908 (120601 (119887) sdot 1198861015840

119886)

(103)





lowast with bracket

[(1198871 1198711 1205731) (1198872 1198712 1205732)] = (0 1198871 [log 120585 1198712]

minus 1198872 [log 120585 1198711] + [1198711 1198712] ⟨[log 120585 1198711] 1198712⟩)(104)


⟨(1198871 1198711 1205731) (1198872 1198712 1205732)⟩

= 12057311198872 +12057321198871 + res (11987111198712) (105)




120590)



120590oplus INT

120590) in which

DO = 120572119888 +

119899

sum

119894=0119886119894120585119894

| 119886119894isinA 120572 isinR


119894=minusinfin

119886119894120585119894

| 119886119894isinA 120573 isinR

(106)



(Ψ119863119874119863119874 119868119873119879)

(Ψ120590119863119874119863119874

120590 119868119873119879

120590)

(107)

are Manin triples




119899DO = Ψ


119894



Ψ119899DO = INT

119894oplus DO

119894 (108)

where

INT119894=

120573 log 120585119894+

(minus1)sum


119886120579120585120579

| 119886120579isin119860

DO119894=

120572119888119894+

(119899)

sum

(0)isinΓ119899

119886120579120585120579

| 119886120579isin119860

(109)



119894 119863119874

119894) form a Ma-






decomposition

Ψ119899DO = Ψ




[(119860 120572 120573) (119861 120572 120573)] = ([119860 119861] + [120573 sdot log 120585 119860]

minus [120573 sdot log 120585 119861] 119888 (119860 119861) 0) (111)

and bilinear form

⟨119860 119861⟩ = res (119860 119861) + 120572 sdot 120573 +120573 sdot 120572 (112)


119899) 0 = (0 0) isin R119899 and 119888(119860 119861) = (1198881(119860 119861)





⟨]120575119865

120575120583⟩ =

119889

119889120598

10038161003816100381610038161003816100381610038161003816120598=0119865 (120583 + 120598]) (113)



119865 119866 (120583) = ⟨120583 [120575119865

120575120583120575119866

120575120583]⟩ (114)



119889120583

119889119905= ⟨120583 [sdot

120575119867

120575120583]⟩ (115)


119877on g by

[119875 119876]119877= [119877 (119875) 119876] + [119875 119877 (119876)] 119876 119875 isin g (116)





+minus



plusmn We write

119875plusmn= 120587


119877= [119875

+ 119876

+]minus[119875

minus 119876

minus] and

119865 119866119877= ⟨120583 [

120575119865

120575120583120575119866

120575120583]

119877

⟩ (117)


⟨120583 [120575119865

120575120583 sdot]⟩ = 0 (118)


119865 119866119877= ⟨120583 [(

120575119865

120575120583)

+

120575119866

120575120583]⟩ (119)


119883119867(120583) sdot 119865 = ⟨120583 [minus(

120575119867

120575120583)

+

120575119865

120575120583]⟩ (120)


⟨119875 [119876 119877]⟩ = ⟨[119877 119875] 119876⟩ (121)


119889119875

119889119905= [(

120575119867

120575120583)

+

119875] (122)



⟨[119875 119876] 119878⟩ = ⟨[119878 119875] 119876⟩ 119878 119875 119876 isin ΨDO (123)


minusoplus ΨDO

+allows us to




Ψ119863119874 Then

120575119867119896

120575119871= 119896119871

119896minus1 (124)


119867119896 119867

119897119877(119871) = 0 for any 119877-matrix 119877




119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (125)


120590 DO

120590 and INT

120590


120590as



120590 The


119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (126)



119899DO DO

119899 and INT

119899allows us


[ ]119877 Once more we set 119867

119896(119871) = res(119871119896) for 119871 isin Ψ

119899DO


120597119871

120597119905= [(119896119871

119896minus1)+

119871] (127)




⟨(119860 120572) (119861 120573)⟩ = ⟨119860 119861⟩ + 120572120573 = res (119860119861) + 120572120573 (128)




⟨120597119871

120597119905 119876⟩ = ⟨[

120597119867

120597119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (129)

119889119897

119889119905= 0 (130)



119889 ( )

119889119905sdot (119876 119903) = ⟨( ) [(119876 119903)

120575119867

120575 ( )]⟩

= ⟨( ) [(119876 119903) (120575119867

120575120575119867

120575)]⟩

= ⟨( ) ([119876120575119867

120575] 119888 (119876

120575119867

120575))⟩

(131)


⟨120597 (119871 119897)

120597119905 (119876 119903)⟩

= ⟨(119871 119897) ([119876120575119867

120575119871] 119888 (119876

120575119867

120575119871))⟩

= ⟨119871 [119876120575119867

120575119871]⟩+ 119897119888 (119876

120575119867

120575119871)

(132)


⟨120597119871

120597119905 119876⟩+ 119903

120597119897

120597119905= ⟨[

120575119867

120575119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (133)





res ([log 120585 119886120585] 119887120585) = res (119887120585 [log 120585 119886120585])

= res(119887120585infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 1205851minus119895)

= res(infin

sum

119895=0

(minus1)119895+1

119895(119887120575

119895

(119886) 1205852minus119895

+ 119887120575119895+1

(119886) 1205851minus119895))

=(minus1)3+1

3119887120575

3(119886) +

(minus1)2+1

2119887120575

3(119886)

= minus16120591 (119887120575

3(119886))

(134)


⟨(119886120585 120572) (119887120585 120573)⟩ = 120591 (119886119887) + 120572120573 (135)


⟨119889119906

119889119905120585 119887120585⟩ = ⟨[ℎ (119906) 120585 119906120585] 119887120585⟩

minus16119897120591 (119887120575

3(ℎ (119906)))

(136)


119889119906

119889119905= ℎ (119906) 120575 (119906) minus 119906120575 (ℎ (119906)) minus (

16) 119897120575

3(ℎ (119906)) (137)




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119899DO = Ψ


119894



Ψ119899DO = INT

119894oplus DO

119894 (108)

where

INT119894=

120573 log 120585119894+

(minus1)sum


119886120579120585120579

| 119886120579isin119860

DO119894=

120572119888119894+

(119899)

sum

(0)isinΓ119899

119886120579120585120579

| 119886120579isin119860

(109)



119894 119863119874

119894) form a Ma-






decomposition

Ψ119899DO = Ψ




[(119860 120572 120573) (119861 120572 120573)] = ([119860 119861] + [120573 sdot log 120585 119860]

minus [120573 sdot log 120585 119861] 119888 (119860 119861) 0) (111)

and bilinear form

⟨119860 119861⟩ = res (119860 119861) + 120572 sdot 120573 +120573 sdot 120572 (112)


119899) 0 = (0 0) isin R119899 and 119888(119860 119861) = (1198881(119860 119861)





⟨]120575119865

120575120583⟩ =

119889

119889120598

10038161003816100381610038161003816100381610038161003816120598=0119865 (120583 + 120598]) (113)



119865 119866 (120583) = ⟨120583 [120575119865

120575120583120575119866

120575120583]⟩ (114)



119889120583

119889119905= ⟨120583 [sdot

120575119867

120575120583]⟩ (115)


119877on g by

[119875 119876]119877= [119877 (119875) 119876] + [119875 119877 (119876)] 119876 119875 isin g (116)





+minus



plusmn We write

119875plusmn= 120587


119877= [119875

+ 119876

+]minus[119875

minus 119876

minus] and

119865 119866119877= ⟨120583 [

120575119865

120575120583120575119866

120575120583]

119877

⟩ (117)


⟨120583 [120575119865

120575120583 sdot]⟩ = 0 (118)


119865 119866119877= ⟨120583 [(

120575119865

120575120583)

+

120575119866

120575120583]⟩ (119)


119883119867(120583) sdot 119865 = ⟨120583 [minus(

120575119867

120575120583)

+

120575119865

120575120583]⟩ (120)


⟨119875 [119876 119877]⟩ = ⟨[119877 119875] 119876⟩ (121)


119889119875

119889119905= [(

120575119867

120575120583)

+

119875] (122)



⟨[119875 119876] 119878⟩ = ⟨[119878 119875] 119876⟩ 119878 119875 119876 isin ΨDO (123)


minusoplus ΨDO

+allows us to




Ψ119863119874 Then

120575119867119896

120575119871= 119896119871

119896minus1 (124)


119867119896 119867

119897119877(119871) = 0 for any 119877-matrix 119877




119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (125)


120590 DO

120590 and INT

120590


120590as



120590 The


119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (126)



119899DO DO

119899 and INT

119899allows us


[ ]119877 Once more we set 119867

119896(119871) = res(119871119896) for 119871 isin Ψ

119899DO


120597119871

120597119905= [(119896119871

119896minus1)+

119871] (127)




⟨(119860 120572) (119861 120573)⟩ = ⟨119860 119861⟩ + 120572120573 = res (119860119861) + 120572120573 (128)




⟨120597119871

120597119905 119876⟩ = ⟨[

120597119867

120597119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (129)

119889119897

119889119905= 0 (130)



119889 ( )

119889119905sdot (119876 119903) = ⟨( ) [(119876 119903)

120575119867

120575 ( )]⟩

= ⟨( ) [(119876 119903) (120575119867

120575120575119867

120575)]⟩

= ⟨( ) ([119876120575119867

120575] 119888 (119876

120575119867

120575))⟩

(131)


⟨120597 (119871 119897)

120597119905 (119876 119903)⟩

= ⟨(119871 119897) ([119876120575119867

120575119871] 119888 (119876

120575119867

120575119871))⟩

= ⟨119871 [119876120575119867

120575119871]⟩+ 119897119888 (119876

120575119867

120575119871)

(132)


⟨120597119871

120597119905 119876⟩+ 119903

120597119897

120597119905= ⟨[

120575119867

120575119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (133)





res ([log 120585 119886120585] 119887120585) = res (119887120585 [log 120585 119886120585])

= res(119887120585infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 1205851minus119895)

= res(infin

sum

119895=0

(minus1)119895+1

119895(119887120575

119895

(119886) 1205852minus119895

+ 119887120575119895+1

(119886) 1205851minus119895))

=(minus1)3+1

3119887120575

3(119886) +

(minus1)2+1

2119887120575

3(119886)

= minus16120591 (119887120575

3(119886))

(134)


⟨(119886120585 120572) (119887120585 120573)⟩ = 120591 (119886119887) + 120572120573 (135)


⟨119889119906

119889119905120585 119887120585⟩ = ⟨[ℎ (119906) 120585 119906120585] 119887120585⟩

minus16119897120591 (119887120575

3(ℎ (119906)))

(136)


119889119906

119889119905= ℎ (119906) 120575 (119906) minus 119906120575 (ℎ (119906)) minus (

16) 119897120575

3(ℎ (119906)) (137)




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











⟨[119875 119876] 119878⟩ = ⟨[119878 119875] 119876⟩ 119878 119875 119876 isin ΨDO (123)


minusoplus ΨDO

+allows us to




Ψ119863119874 Then

120575119867119896

120575119871= 119896119871

119896minus1 (124)


119867119896 119867

119897119877(119871) = 0 for any 119877-matrix 119877




119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (125)


120590 DO

120590 and INT

120590


120590as



120590 The


119877is

120597119871

120597119905= [(119896119871

119896minus1)+

119871] (126)



119899DO DO

119899 and INT

119899allows us


[ ]119877 Once more we set 119867

119896(119871) = res(119871119896) for 119871 isin Ψ

119899DO


120597119871

120597119905= [(119896119871

119896minus1)+

119871] (127)




⟨(119860 120572) (119861 120573)⟩ = ⟨119860 119861⟩ + 120572120573 = res (119860119861) + 120572120573 (128)




⟨120597119871

120597119905 119876⟩ = ⟨[

120597119867

120597119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (129)

119889119897

119889119905= 0 (130)



119889 ( )

119889119905sdot (119876 119903) = ⟨( ) [(119876 119903)

120575119867

120575 ( )]⟩

= ⟨( ) [(119876 119903) (120575119867

120575120575119867

120575)]⟩

= ⟨( ) ([119876120575119867

120575] 119888 (119876

120575119867

120575))⟩

(131)


⟨120597 (119871 119897)

120597119905 (119876 119903)⟩

= ⟨(119871 119897) ([119876120575119867

120575119871] 119888 (119876

120575119867

120575119871))⟩

= ⟨119871 [119876120575119867

120575119871]⟩+ 119897119888 (119876

120575119867

120575119871)

(132)


⟨120597119871

120597119905 119876⟩+ 119903

120597119897

120597119905= ⟨[

120575119867

120575119871 119871] 119876⟩+ 119897119888 (119876

120575119867

120575119871) (133)





res ([log 120585 119886120585] 119887120585) = res (119887120585 [log 120585 119886120585])

= res(119887120585infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 1205851minus119895)

= res(infin

sum

119895=0

(minus1)119895+1

119895(119887120575

119895

(119886) 1205852minus119895

+ 119887120575119895+1

(119886) 1205851minus119895))

=(minus1)3+1

3119887120575

3(119886) +

(minus1)2+1

2119887120575

3(119886)

= minus16120591 (119887120575

3(119886))

(134)


⟨(119886120585 120572) (119887120585 120573)⟩ = 120591 (119886119887) + 120572120573 (135)


⟨119889119906

119889119905120585 119887120585⟩ = ⟨[ℎ (119906) 120585 119906120585] 119887120585⟩

minus16119897120591 (119887120575

3(ℎ (119906)))

(136)


119889119906

119889119905= ℎ (119906) 120575 (119906) minus 119906120575 (ℎ (119906)) minus (

16) 119897120575

3(ℎ (119906)) (137)




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











res ([log 120585 119886120585] 119887120585) = res (119887120585 [log 120585 119886120585])

= res(119887120585infin

sum

119895=1

(minus1)119895+1

119895120575119895

(119886) 1205851minus119895)

= res(infin

sum

119895=0

(minus1)119895+1

119895(119887120575

119895

(119886) 1205852minus119895

+ 119887120575119895+1

(119886) 1205851minus119895))

=(minus1)3+1

3119887120575

3(119886) +

(minus1)2+1

2119887120575

3(119886)

= minus16120591 (119887120575

3(119886))

(134)


⟨(119886120585 120572) (119887120585 120573)⟩ = 120591 (119886119887) + 120572120573 (135)


⟨119889119906

119889119905120585 119887120585⟩ = ⟨[ℎ (119906) 120585 119906120585] 119887120585⟩

minus16119897120591 (119887120575

3(ℎ (119906)))

(136)


119889119906

119889119905= ℎ (119906) 120575 (119906) minus 119906120575 (ℎ (119906)) minus (

16) 119897120575

3(ℎ (119906)) (137)




Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Formal Pseudodifferential Operators in One and Several Variables ... · 2019. 7....

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Transcript of Research Article Formal Pseudodifferential Operators in One and Several Variables ... · 2019. 7....