Poisson Ore extensions

8/12/2019 Poisson Ore extensions

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Poisson Ore extensions

N. CICCOLI 11 2, Dipartimento di Matematica e Informatica (Universita diPerugia), Via Vanvitelli 1 - 06123 Perugia - Italy

Abstract. In this paper we describe the Poissongeometrical coun-

terpart of algebraic Ore extensions in cohomological language. This

allows us to adress easily the problem of unimodularity of Poissonextensions.

1 Introduction

An Ore extension of a given algebra is a non commutative algebra obtainedfrom the starting one by adding a generator and a set of relations. Such alge-bras are also known as skew polynomial rings and their origins could be tracedback to Hilbert. With such a long history on their backs, and containing signifi-cant examples as the Weyl algebras, it is no surprise theyve been much studied

throughout the twentieth century.Since the appearence of quantum groups and quantum spaces this attention

was revived. In fact many significant examples of quantum algebras can beconstructed via a chain of iterated Ore extensions (e.g. [?]). This was usedextensively to obtain results on the algebraic properties of such algebras.

1E-mail: [email protected]; supported by Gnsaga, Prin05 Geometria Riemanniana

e strutture differenziabili, also Transfer of Knowledge Nocommutative geometry and quantum

groups, contract no. MKTD-CT-2004-509794 This paper is in final form and no version of it

will appear elsewhere,2Conferenza tenuta il 30 Aprile 2008


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The word quantumhere refers to the fact that these algebras can be seenas deformation of the algebra of functions on a classical phase space, i.e. acommutative Poisson algebra. It was then reasonable to seek for an analogue ofOre extensions in the context of Poisson algebras.

These notes grew out on an attempt to understand in more geometrical termsa construction given by [?]. In the cited paper this author solves the problem offinding a semiclassical analogue of skew polynomial rings, by defining a version ofskew Poisson polynomial rings. We recast his construction in more geometrical(i.e. global) terms, showing a possible use of Poisson cohomology with twistedcoefficients.

This has to be considered as a preliminary step in developing a thoroughgeometric theory of Poisson Ore extensions, which should allow to understandhow to relate relevant geometrical invariants of the original Poisson manifold andof the extended one.

2 Poisson Ore extensions

Let (M, ) be a Poisson manifold. We will denote with d the usual Poissoncohomology coboundary on multivector fields d = [, ] : X

k(M) Xk+1(M).Let be a Poisson vector field, i.e. X(M), d= 0.

We can then define

d, = [, ] + : Xk(M) Xk+1(M)

Proposition 2.1. The operatord, has square zero.

Proof. This is a 1-line computation:

d2,P = [, [, P] ] + [, P] + [, P] + P = [, P] + [, P]

where we have used properties of the Schouten bracket and [, ] = [, ] =

0.

The cohomology operator defined above is, in fact, quite general. ConsiderM R as a trivial line bundle on M. Then Poisson vector fields on M arein 1-1 correspondence with flat contravariant connections on M R ([?], orPoisson line bundles in the terminology of [?]) associated to it. The cohomologyofd, is just the Poisson cohomology with coefficients in this connection. Thiscohomology, more correctly depends not on but rather on its cohomology class[] H1(M). In the special case in which Mis an orientable Poisson manifoldandMR is identified with the canonical line bundlenTMvia a volume formthe corresponding cohomologies where considered also in [?], (see question 3 at


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the end of his paper). We will denote such cohomology groups withH,(M),and sometime refer to them as twisted Poisson cohomology.

We will call any pair (, ) such that is a Poisson vector field on Mandisa 1cocycle intwisted Poisson cohomology a PoissonOre dataThe followingis the geometrical restatement of theorem 1.2 of [?].

Proposition 2.2. Let us considerMR and denote witht the coordinate alongR. Let, X(M). Then = + (t + ) t is a Poisson bivector onMRif and only if(, ) is a PoissonOre data.

Proof. Again a very neat, coordinate-free, computation shows that

[, ] = [, ] + 2(d, t) .

The right hand side is zero iff both summands are zero ([ , ] gives no contribu-tion in t) hence the thesis.

To compare with the cited theorem just remark that is the bivector corre-sponding to brackets given by formulae (1.1) and d,= 0 is exactly condition(1.2) of [?].

The case in which is the modular vector field and = 0 appeared recentlyin the context of deformation quantization in [?], playing a relevant role in ex-plaining the Poincare duality for formal quantizations of unimodular Poissonmanifolds. Current work is going on to explain how to extend such results toglobal (i.e. with fixed deformation parameter value) quantizations ([?]).

Proposition 2.3. Let1 and2 be cohomologous cocycles inH1,(M). Then

(1; 1) (1; 2) .

Proof. The vector fields 1 and 2 are cohomologous iff there exists f C(M)

such that 1=2+f . It is then easily checked that changing the line variablefrom t to s= t+fgives the desired isomorphism.

Proposition 2.4. Let [1] = [2] H1(M), say 1 = 2+Xf. Let 1 be a

d,11cocycle. Then2 = ef1 is ad,21cocycle and

(1; 1) (2; 2) .

Proof. The first statement follows from

d,2(ef1) =d(e

f1) + ef(1 Xf) 1=d(e

f) 1 + efd,11 e

fXf 1 = 0

For the second just consider the change of variables xefx.


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1. Should one put himself in the context of smooth algebraic affine Pois-son varieties, where everything we did up to now still works, some careis needed on this last statement. Instead of having Poisson vector fieldsdiffering by an Hamiltonian one, one should consider them differing by alogHamiltonian vector field. Further comments on log-Hamiltonian vectorfields can be found in the appendix at [?].

2. What about a possible higher dimensional analogue of this situation? Itwould be natural to ask the following: given a flat contravariant connectiononMRn (i.e. annn-matrix of Poisson vector fields) and a corresponding

twisted Poisson 1-cocycle (i.e. a vector ofn vector fields on M such thatdi=

jijj) is it possible to construct a Poisson structure onMR

n.Unfortunately the answer is not positive in general. A problem which isalready known for Ore extensions where one needs additional condition. Inthe recent paper [?] some sufficient conditions for the existence of algebraicdouble Ore extensions are given. It would be interesting to translate theminto the Poisson theory and check whether they can be interpretated in thelanguage above.

3. Let us consider the case in which M is symplectic. The sharp map thendefines an isomorphism of complexes, and thus descends to an isomorphism

in cohomology H(M) HdeR(M). Similarly, as explained in appendix1, the twisted Poisson cohomology is isomorphic to the twisted de Rhamcohomology, and therefore, in the trivial line bundle case, again to the deRham cohomology itself. The Poisson structure is therefore defined, upto isomorphism, by the two de Rham classes [()] and [()].

3 Unimodularity

As an example of application of this global language let us describe the

problem of unimodularity of this Poisson extension. Let us recall that fora given orientable Poisson manifold with a volume form the modularvector field is defined as the vector field such that

L= .

The manifold is said to be unimodular if the Poisson cohomology class of is zero.

Proposition 3.1. Let (M, ) be a Poisson manifold, let be a Poissonvector field and let X1(M) be an Poisson vector field. Let be a


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volume form onM and is the corresponding modular vector field; define

functionsf andf as:

L=f L= f

(1)

The modular vector field of(M; ;) is( ) + (tf +f)t.

Proof. Let be a volume form on M. ComputeL( dt) with the usualmultivector fields formulae: then

L( dt) =() dt+ (tL+L)

hence the thesis.

From this proposition it is obvious that if we choose equal to the modularvector field of and= 0 any Poisson manifold can be canonically embed-ded into a unimodular Poisson manifold with codimension 1, we will callthis the canonical unimodular Poisson Ore extension. More generally anunimodular Poisson Ore extension can be found whenever = (whichis also a necessary condition) and f = 0; i.e. if we are able to choose a invariant volume form.

One more remark, we believe, is of some interest. Let = and be a 1-cocycle. Therefore we trivially have: i[,]+ = 0. A direct computationshows that this implies L(i) = 0. Therefore i is a Poisson homologyclass in Hn1(M, ) this homology class does not depend on the volumeform, but just on the Ore data. If, in fact, we change the volume form tog,say withgeverywhere positive, we have a modular vector field =+Xlngand, correspondingly (as in proposition ??) a 1-cycle = g1, so thatig =i.

Let us show this concretely for an example. Let be the quadratic Poissonstructure on R2 given by

{x, y}=x2 +y2

2

(the normalization factor 2 is used to simplify computations). An easycomputation with the volume form dx dy shows that the modular vectorfield is the rotation generator = xy yx. Let us then consider thestandard Poisson line bundle corresponding to it. A direct computation(along the same lines of [?]; additional details are given in Appendix B)shows that

H0(R

2

; ) = 0 H1(R

2

; ) =1, xx yy, yx+xy


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H2(R2; ) =1,x,y,xy,x2 y2x y

In particular both1 = xxyyand 2=yx+xyare 1cocycles underwhich the standard volume form is invariant (i.e. the correspondingf is0). They allow to define the following unimodular Poisson Ore extensionsof the singular quadratic Poisson structure on R3 (with coordinates x, y,tto be consistent with previous notations).

(; 1) =x2 +y2

2 x y+ (yt+x)x t + (tx y)y t, (2)

(; 2) =x2 +y2

2 x y+ y(1 t)x t +x(t+ 1)y t. (3)

4 Final remarks

Weve introduced in global geometric terms the notion of skew Poissonextension. Many known examples of Poisson manifolds arising in the theoryof PoissonLie groups and their homogeneous spaces fits into this family,just like their corresponding quantum analogues.

The theory of skew polynomial extensions has been much developed from

an algebraic point of view. We do believe that many interesting cohomo-logical Poisson invariants could be computed either directly from algebraictechniques and good quantization theorems (like formality theorems forchains, see [?]) or by translating such techniques (e.g. results in [?]) di-rectly into the Poisson world.

A Poisson vector bundles

Let E be a vector bundle on a manifold M; it is called a Poisson vector

bundle if there exists a bracket

{, }E :C(M) (E) (E)

such that:

{f g , s}E = f{g, s}E +g{f, s}E, f, g C(M), s (E)

{{f, g}, s}E = {f, {g, s}E}E {g, {f, s}E}E, f, g C(M), s (E)

{f,gs}E = {f, g}s+g{f, s}E, f, g C(M), s (E) .

IfEis a line bundle with a trivializing section 1then f {f, 1}E defines

a Poisson vector field on M.


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Remark: This definition can be reinterpreted by remarking that thebracket{f, s}E does depend only form df; therefore we could as well con-sider the bracket {, }E as a map

: 1M (E) (E)

asC(M)-linear extension of(df, s) ={f, s}E. We will use the notation(, s) =s. In this notation the defining identities become

(f s) = fs+ (()f)s ;

[,]s = [, ]s .

This explains the interpretation of Poisson vector bundles as flat contravari-ant connections. From this point of view the Poisson cohomology withcoefficients in a Poisson vector bundle generalizes the cohomology definedon a bundle with a flat connection (see [?], I.7).

B Twisted Poisson cohomology

In this appendix we sketch the details of the twisted Poisson cohomologycomputations lying behind the example of the singular quadratic Poissonstructure on the plane.

Let us rewrite the quadratic Poisson structure on the plane in complexcoordinates as 0 = zzz z. The corresponding modular vector field isthen: = zz zz. It is easily shown by direct computation that:

d,f= (zzzf+zf)z+ (zzzf+ zf)z (4)

for any f C(C) and, ifX=z+ z

d,X=zz(z z)z z . (5)

The same argument used in [?] can be used here to reduce computationsto algebraic coefficients; we will therefor limit ourselves to polynomials inz,z.

Let us denote with Vi the space of homogeneous polynomials in z, z ofdegree i. Then d, as an operator on functions splits into a family ofmaps i : Vi1 Viz Viz and as an operator on vector fields splits asi : Viz Viz Vi+1z z. It is easily checked that whenever i 1,i is injective and i is surjective. This, added to a simple dimension

count, guarantees that no contributions on cohomology may arise from


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terms whose coefficients are polynomials of degree higher than 2. Now wejust need to check the contributions coming from lower order maps. Checkthat:

0(1) =, 0(1) = 0

and

1zz =zzz z 1zz = 0

1zz = 0 1zz =zzz z

From these equalities the results claimed above easily follows.

References

[1] R. Bott and J.L. Tu, Differential forms in algebraic topology, Springer-Verlag New York, 1982.

[2] N. Ciccoli, Poisson homology of Poisson-Ore extensions, in prepara-tion.

[3] N. Ciccoli and U. Khramer, Poincare duality for global quantizations,

in preparation.[4] V. Dolgushev,The van den Bergh duality and the modular symmetry

of a Poisson variety, math.QA/0612288.

[5] V. Dolgushev,A formality theorem for Hochschild chains,Adv. Math.200, 51-101 (2006).

[6] R.L. Fernandes, Connections in Poisson geometry I. Holonomy andinvariants,Journ. Diff. Geom. 54, 303366 (2000).

[7] V. Ginzburg, Momentum mappings and Poisson cohomology, Int. J.Math. 7, 329358 (1996).

[8] V. Ginzburg,Grothendieck groups of Poisson vector bundles, J. Sympl.Geom. 1, 121169 (2002).

[9] J.A. Guccione and J.J. Guccione, Hochschild and cyclic homology ofOre extensions and some examples of quantum algebras, K-theory 12,259267 (1997).

[10] D.Jordan, Iterated skew polynomial rings and quantum groups, J. Al-gebra156, 194218 (1993).

[11] S.-Q. Oh,Poisson polynomial rings,Comm. in Algebra 34, 12651277(2006).


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[12] P. Xu, Gerstenhaber algebras and BV algebras in Poisson geometry,Comm. Math. Phys. 200, 545560 (1999).

[13] J.J. Zhang and J. Zhang,Double Ore extensions, arXiv:0712.2549.

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