Lie Algebroids

8/10/2019 Lie Algebroids

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HOMOLOGY AND MODULAR CLASSES OF LIE ALGEBROIDS

JANUSZ GRABOWSKI, GIUSEPPE MARMO, PETER W. MICHOR

Abstract. For a Lie algebroid, divergences chosen in a classical way leadto a uniquely defined homology theory. They define also, in a natural way,

modular classes of certain Lie algebroid morphisms. This approach, appliedfor the anchor map, recovers the concept of modular class due to S. Evens,J.-H. Lu, and A. Weinstein.

1. Introduction

Homology of a Lie algebroid structure on a vector bundle EoverMare usuallyconsidered as homology of the corresponding Batalin-Vilkovisky algebra associatedwith a chosen generating operator for the Schouten-Nijenhuis bracket on multi-sections ofE. The generating operators that are homology operators, i.e. 2 = 0,can be identified with flat E-connections on

top E (see [17]) or divergence op-erators (flat right E-connections on M R, see [8]). The problem is that thehomology group depends on the choice of the generating operator (flat connection,divergence) and no one seems to be privileged. For instance, if a Lie algebroid onTMassociated with a Poisson tensor P on M is concerned, then the traditionalPoisson homology is defined in terms of the Koszul-Brylinski homology operatorP = [d, iP]. However, the Poisson homology groups may differ from the homology

groups obtained by means of 1-densities on M. The celebrated modular class ofthe Poisson structure [18] measures this difference. Analogous statement is validfor triangular Lie bialgebroids [11].

The concept of a Lie algebroid divergence, so a generating operator, associatedwith a volume form, i.e. nowhere-vanishing section of

top E, is completelyclassical (see [11], [17]). Less-known seems to be the fact that we can use odd-forms instead of forms (cf. [2]) with same formulas for divergence and that suchnowhere-vanishing volume odd-forms always exist. The point is that the homologygroups obtained in this way are all isomorphic, independently on the choice ofthe volume odd-form. This makes the homology of a Lie algebroid a well-definednotion. From this point of view the Poisson homology is not the homology of theassociated Lie algebroid TMbut a deformed version of the latter, exactly as theexterior differentiald= d +of Witten[19]is a deformation of the standardde Rham differential.

Date: March 24, 2005 .

2000 Mathematics Subject Classification. Primary 17B56, 17B66, 17B70; Secondary 53C05 .Key words and phrases. Lie algebroid, de Rham cohomology, Poincare duality, divergence.The research of J. Grabowski supported by the Polish Ministry of Scientific Research and

Information Technology under the grant No. 2 P03A 020 24. The research of P. Michor wassupported by FWF Projects P 14195 and P 17108.

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2 JA NUS Z G RABOWSKI, GIUSE PP E MA RMO, PETER W. MICH OR

In this language, the modular class of a Lie algebroid morphism : E1 E2covering the identity onMis defined as the class of the difference between the pull-back of a divergence on E

2and a divergence on E

1, both associated with volume

odd-forms. In the case when : E T M is the anchor map, we recognize thestandard modular class of a Lie algebroid [3] but it is clear that other (canonical)morphisms will lead to other (canonical) modular classes.

2. Divergences and generating operators

2.1. Lie algebroids and their cohomology. Let :E Mbe a vector bundle.LetAi(E) = Sec(

i E) fori = 0, 1, 2, . . . , letAi(E) = {0}for i


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HOMOL OGY AND MOD ULAR CLASSES OF LI E ALGEBROIDS 3

algebraA(E) which is completely determined by its restriction to Sec(E). In fact,it is easy to see (cf. [8]) that

(3) (X1 Xn) =

i

(1)i+1(Xi)X1 . . . i Xn+

+k


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(a) =

ki(k)Xka where the Xk and

k are dual local frames for E andE, respectively. Since

2 =k,j

i(j )Xj i(k)Xk =k,j

i(j )i(k)(XjXk XjXk),

2 = 0 is equivalent to

(8)j,k

i(j )i(k)R(Xj , Xk) = 0,

whereRis the curvature tensor of. For a Levi-Civita connection the generatingoperator is really a homology operator due to the following lemma.

Lemma 2.3. A torsionfree connection onEsatisfies simultaneously the Bianchiand the Ricci identity if and only if(8)holds for dual local framesXk andk ofEandE, respectively.

Proof. (8) is equivalent to

j,kR(Xj , Xk)(k j ) = 0 for all forms . Itsuffices to check this for a function or a 1-form due to the derivation property ofcontractions. For a function f we havej,k

R(Xj , Xk)(f k j ) =

j,k

f

R(Xj , Xk)(k) j + k R(Xj , Xk)

(j )

= 2fs,j,k

Rkjkss j

and this vanishes for allfif and only ifRkjks is symmetric in (j, s), i.e., if the Ricci

identity holds. For a 1-form, say i, we have

j,k

R(Xj , Xk)(k j i) =

=j,k

R(Xj , Xk)

(k j ) i + k j R(Xj , Xk)(i)

=

= 0 +j,k,s

Rijksk j s

and this vanishes for all i if and only if

cycl(j,k,s) Rijks = 0, i.e., if the first Bianchi

identity holds.

Corollary 2.4. Any Levi-Civita connection for a Riemannian metric on a LiealgebroidE induces a flat connection on

top E, thus also ontop E.

3. Homology of the Lie algebroid3.1. Getting divergences from odd forms. There is no distinguished diver-gence for the Lie algebroid structure on E, but there is a distinguished subset ofdivergences which we may obtain in a classical way. Firstly, suppose that the linebundle

top E is trivializable. So we can choose a vector volume, i.e., a nowherevanishing section Sec(

top E). Then the formula(9) LX = div(X), whereXSec(E)


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defines a divergence div. We observe that div = div. Thus for the non-orientable case we look for sections of a bundle over Mwhich locally consists of non-ordered pairs {

,

} for an open coverM=

U

such that the sets {

,

}

and{, }coincide when restricted to U U . The fundamental observationis that such global sections always exist and define global divergences. This isbecause they can be viewed as sections of the bundle | Vol |E = (

top E)0/Z2,where (

topE)0 is the bundle

topE with the zero section removed and divided

by the obvious Z2-action of passing to the opposite vector. The bundle | Vol |Eis an 1-dimensional affine bundle modelled on the vector bundle M R, and alsoa principal R bundle where t R acts by scalar multiplication with et. Since ithas a contractible fiber, sections always exist. Note that sections ||of| Vol |E areparticular cases of odd forms, [2]: Letp : MMbe the two-fold covering ofMonwhich pEis oriented, namely the set of vectors of length 1 in the line bundle overMwith cocycle of transition functions sign det(), where :U U GL(V)is the cocycle of transition functions for the vector bundleE. Then the odd forms

are those forms onp

Ewhich are in the1 eigenspace of the natural vector bundleisomorphism which covers the decktransformation ofM. So odd forms are certainsections of a line bundle over a two-fold covering of the base manifold M. Thisis related but complementary to the construction of the line bundle (over M) ofdensities which involve the cocycle of transition functions | det()|. For example,any Riemannian metric g on the vector bundle E induces an odd volume form||g Sec(| Vol |E) Sec(| Vol |E) which locally is represented by the wedgeproduct of any orthonormal basis of local sections ofE (thus E). Note that suchproduct is independent on the choice of the basis modulo sign, so our odd volumeis well defined.

For the definition of a divergence div || associated to || Sec(| Vol |E) we willwrite simply

(10) LX ||= div||(X)|| forXSec(E).

Note that the distinguished set Div0(E) of divergences obtained in this way fromsections of| Vol |Ecorresponds (in the sense of (6)) to the set of those flat connec-

tions ontop E whose holonomy group equals Z2: Associate the horizontal leaf

|| to such a connection, and note that a positive multiple of || gives rise to thesame divergence.

In the case of a vector bundle Riemannian metric g on E a natural questionarises about the relation between the divergence div||g associated with the oddvolume ||g induced by the metric g and the divergence divg induced by the flat

Levi-Civita connection g ontop E top E.

Theorem 3.2. For any vector bundle Riemannian metricg onE

div||g = divg.

Proof. Let X1, . . . , X n be an orthonormal basis of local sections of E andk = g(Xk, ) be the dual basis of local sections of E

, so that ||g is locally


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represented by1 . . . n. For any local section XofE

div||g (X) = LX (1 n), X1 Xn=

1 n, LX (X1 Xn)=

k

k, [X, Xk]=

k

k, XXk XkX=

k

g(Xk, X Xk)

k

i(k)XkX.

But

ki(k)XkX= divg (X) and

2

k

g(Xk, X Xk) =

k

(X)g(Xk, Xk)

k

X (g)(Xk, Xk) = 0,

where : E T Mis the anchor of the Lie algebroid on E, since is Levi-Civity(g= 0).

3.3. The generating operator for an odd form. The corresponding generatingoperator || for the divergence of a non-vanishing odd form || can be definedexplicitly by

La||=i(||(a))||,

whereLa =ia d (1)|a|d ia is the Lie differential associated with a A

|a|(E) sothat

(11) i(||(a))||= (1)|a|d ia||.

In other words, locally over Uwe have

(12) ||(a) = (1)|a| 1 d (a)

where is the isomorphism ofA(E)|U and A(E)|U given by (a) = ia, for a

representative of ||. Note that the right hand side of (12) depends only on ||and not on the choice of the representative since d = d . Formula (12)gives immediately2||= 0, which also follows from the remark on flat connections

above. So || is a homology operator.Moreover, it is also a generating operator. Namely, using standard calculus of

Lie derivatives we get

Lab = ib La (1)|a|i[a,b]+ (1)

|a| |b|iaLb

which can be rewritten in the form

(13) i[a,b]= (1)|a|

Lab+ ibLa+ (1)|a|(|b|+1)ia Lb

When we apply (13) to|| we get

i[a,b]||= (1)|a|

i(||(a b)) i(||(a) b) (1)|a|i(a ||(b))

||

which proves (2). Thus we get:

Theorem 3.4. For any || Sec(| Vol |E) the formula

(14) La||=i(||(a))||

defines uniquely a generating operator|| Gen(E).


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We remark that formula (14) in the case of trivializabletop

E has been alreadyfound in [11]. In this sense the formula is well known. What is stated in Theorem

3.4is that(14) serves in general, as if the bundletop

E

were trivial, if we replaceordinary forms with odd volume forms.

3.5. Homology of the Lie algebroid. The homology operator of the form ||will be called the homology operator for the Lie algebroid E. The crucial point isthat they all define the same homology. This is due to the fact that |1| and|2|differ by contraction with an exact 1-form.

In general, two divergences differ by contraction with a closed 1-form. Indeed,(div1 div2)(f X) = f(div1 div2)(X), so (div1 div2)(X) = iX for a unique1-form. Moreover, (5) implies thati[X, Y] = [iX, Y] + [X, iY], so is closed.Since both sides are derivations we have

(15) div2 div1 =i.

But for any |1|, |2| Sec(| Vol |E) there exists a positive function F = ef suchthat|2|= F|1|. Then LX |2|= LX (F|1|) =LX (F) |1| +FLX (|1|) so thatdiv|2|(X)|2|= LX (f) |2| + div|1|(X)|2|, i.e., div|2| div|1|= i(df).

To see that the homology of|1| and|2| are the same, note first that |2|=

|1|a + idfa. And then let us gauge A(E) by multiplication with F =ef. This is

an isomorphism of graded vector spaces and we have

ef |1|ef a= |1|a + idfa= |2|a,

so|1| and |2| are graded conjugate operators.This is just the dual picture of the well-known gauging of the de Rham differential

by Witten[19], see also[7] for consequences in the theory of Lie algebroids. Thuswe have proved (cf. [11, p.120]):

Theorem 3.6. All homology operators for a Lie algebroid generate the the samehomology: H(E, |1|) = H(E, |2|). In the case of trivializable

top E, (12)gives Poincare dualityH(E, d)=Htop(E, ||).

3.7. Remark. We got a well-defined Lie algebroid homology, in contrast with thestandard approach when all generating operators are admitted. It is clear thatadding a termiwith a closed 1-from which is not exact, as in (15), will probablychange the homology. But this could be understood as an a priori deformation likein the case of the deformed de Rham differential of Witten [19]:

(16) d= d+ .

Indeed, i(ia) = (1)|a| ia implies i(a) = (1)

|a| e (a) where

e = . Thus we get (1)|a|

1

(d+e) (a) = (|| i)(a), so, at leastin the the trivializable case, there is the Poincare duality

H(E, d + e)=Htop(E, i).

Note that the differentials d appear as part of the Cartan differential calculus forJacobi algebroids, see [10], [6], [7], so that there is a relation between generatingoperators for a Lie algebroid and the Jacobi algebroid structures associated withit.


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4. Modular classes

4.1. The modular class of a morphism. As we have shown, every Lie algebroid

Ehas a distinguished class Div0(E) of divergences obtained from sections of| Vol |E.Such divergences differ by contraction with an exact 1-form. Let now : E1 E2be a morphism of Lie algebroids.

There is the induced map : Div(E2) Div(E1) defined by (div2)(X1) =div2((X1)). The fact that

maps divergences into divergences follows from(f X) = f (X) and the fact that the Lie algebroid morphism respects the anchors,1 = 2 . The space(Div0(E2))Div(E1) consists of divergences which differby insertion of an exact 1-form. Therefore, the cohomology class of the 1-formwhich is defined by the equation

(17) (divE2) divE1 =i, for divEi Div0(Ei), i= 1, 2,

does not depend on the choice of divE1 and divE2 . We will call it the modular classof and denote it by Mod(). Thus we have:

Theorem 4.2. For every Lie algebroid morphism

E1

1

E2

2

M

the cohomology class Mod() = [] H1(E1, dE1) defined by in (17) is welldefined independently of the choice ofdivE1 Div0(E1)anddivE2 Div0(E2).

4.3. The modular class of a Lie algebroid. In the case when the morphism = : E T M is the anchor map of a Lie algebroid E, the modular classMod() is called the modular class of the Lie algebroid E and it is denoted byMod(E). The idea that the modular class is associated with the difference between

the Lie derivative action on

top(E) and on

top TMvia the anchor map is, infact, already present in [3]. Also the interpretation of the modular class as certainsecondary characteristic class of a Lie algebroid, present in [4], is a quite similar. In[4]the trace of the difference of some connections is used instead of the differenceof two divergences. We have

Theorem 4.4. Mod(E) is the modular class Ein the sense of [3].

Proof. The modular class Ein the sense of [3]is defined as the class [] where is given by

(18) LX (a) + a L(X)=X, a

for all sections a oftop

(E) and oftop

(TM), respectively. Let us take |a|

Sec(| Vol |E) and || Sec(| Vol |T M), locally represented by a

Sec(top

(E

|U))and Sec(

top(TM|U)). Let a be a local section of

top Edual to a. ThenLX(a) = div|a|(X) a and LX () =

(div||)(X) so that (18) yields i =(div||) div|a|.

Note that in our approach the modular class Mod(T M) of the canonical LiealgebroidT M is trivial by definition. It is easy to see that the modular class of abase preserving morphism can be expressed in terms of the modular classes of thecorresponding Lie algebroids.


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Theorem 4.5. For a base preserving morphism: E1 E2 of Lie algebroids

Mod() = Mod(E1) (Mod(E2)).

Proof. Letl : El T Mbe the anchor ofEl, l = 1, 2. Take divEl Div0(El),l = 1, 2, and divT M Div0(T M). Since Mod(El) is represented by l, il =divEl

l (divT M) and1 = 2 , we can write

i1 = divE1 1(divT M) =

divE1 (divE2) +

(divE2 2(divT M) = i+ i(2),

where represents Mod(). Thus 1 = + 2.

4.6. The universal Lie algebroid. For any vector bundle : E M thereexists a universal Lie algebroid QD(E) whose sections are the quasi-derivations on

E, i.e., mappings D: Sec(E) Sec(E) such that D(f X) =f D(X) + D(f) X for

fC

(M) andXSec(E), where Dis a vector field on M; see the survey article[5]. Quasi-derivations are known in the literature under various names: covariantdifferential operators[15], module derivations [16], derivative endomorphisms[12],etc. The Lie algebroid QD(E) can be described as the Atiyah algebroid associatedwith the principalGL(n,R)-bundle Fr(E) of frames inE, and quasi-derivations canbe identified with theGL(n,R)-invariant vector fields on Fr(E). The correspondingshort exact Atiyah sequence in this case is

0End(E) QD(E) T M0.

This observation shows that there is a modular class associated to every vector bun-dleE, namely the modular class Mod(QD(E)), which is a vector bundle invariant.

It is also obvious that, viewing a flat E0-connection (representation) in a vectorbundle E over M for a Lie algebroid E0 over M as a Lie algebroid morphism

: E0 QD(E), one can define the modular class Mod().

Question. How isMod(QD(E)) related to other invariants ofE (e.g. character-istic classes)?

4.7. Remark. One can interpret the modular class Mod(E) of the Lie algebroidEas a trace of the adjoint representation. Indeed, if we fix local coordinates u a

onUMa local frameXi of local sections ofEoverU, and the dual framei ofE, then the Lie algebroid structure is encoded in the structure functions

[Xi, Xj ] =

k

ckijXk, (Xi) =

a

ai ua .

Proposition.The modular classMod(E)is locally represented by the closed 1-form

(19) =

i

k

ckik+

a

aiua

i.

Proof. We insert into (18) the elementsa= X1 Xnand= du1 dum.

Since

LXia=

k

ckika and LXi=

a

aiua

,


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10 JANUSZ GRABOWSKI, GIUSEPPE MARMO, PETER W. MICHOR

we get

Xi, a =

k

ckik+

a

aiua

a .

One could say that representing cohomology locally does not make much sense,e.g. the modular class Mod(T M) is trivial so locally trivial. However, rememberthat for a general Lie algebroid the Poincare lemma does not hold: closed formsneed not be locally exact. In particular, for a Lie algebra (with structure constants),(19) says that the modular class is just the trace of the adjoint representation. Inany case, (19) gives us a closed form, which is not obvious on first sight. IfE is atrivial bundle, (19) gives us a globally defined modular class in local coordinates.

4.8. Remark. As we have already mentioned, the modular class of a Lie algebroidis the first characteristic class of R. L. Fernandes [4]. There are also higher classes,shown in[1] to be characteristic classes of the anchor map, interpreted as a repre-sentation up to homotopy. It is interesting if our idea can be adapted to describe

these higher characteristic classes as well.

5. Acknowledgement

The authors are grateful to Marius Crainic and Yvette Kosmann-Schwarzbachfor providing helpful comments on the first version of this note.

References

[1] M. Crainic. Chern characters via connections up to homotopy. arXiv: math.DG/0009229.[2] G. De Rham.Varietes differentiables. Hermann, Paris, 1955.

[3] S. Evens, J.-H. Lu, and A. Weinstein. Transverse measures, the modular class, and a coho-mology pairing for Lie algebroids. Quarterly J. Math., Oxford Ser. (2), 50:417436, 1999.

[4] R. L. Fernandes. Lie algebroids, holonomy and characteristic classes.Adv. Math.170:119179,

2000.

[5] J. Grabowski. Quasi-derivations and Qd-algebroids.Rep. Math. Phys. 52:445451, 2003.[6] J. Grabowski and G. Marmo. Jacobi structures revisited. J. Phys. A: Math. Gen., 34:10975

10990, 2001.[7] J. Grabowski and G. Marmo. The graded Jacobi algebras and (co)homology. J. Phys. A:

Math. Gen., 36:161181, 2003.[8] J. Hubschmann. Lie-Rinehart algebras, Gerstenhaber algebras, and Batalin-vilkovisky alge-

bras. Ann. Inst. Fourier, 48:425440, 1998.[9] J. Hubschmann. Duality for Lie-rinehart algebras and the modular class. J. reine angew.

Math., 510:103159, 1999.[10] D. Iglesias and J.C. Marrero. Generalized Lie bialgebroids and Jacobi structures.J. Geom.

Phys., 40:176199, 2001.

[11] Y. Kosmann-Schwarzbach.Modular vector fields and Batalin-Vilkovisky algebras, in: PoissonGeometry, J. Grabowski and P. Urbanski (eds.), Banach Center Publications 51:109129,

Warszawa 2000.[12] Y. Kosmann-Schwarzbach and K. Mackenzie.Differential operators and actions of Lie alge-

broids. Contemp. Math. 315:213233, 2002.

[13] Y. Kosmann-Schwarzbach and J. Monterde. Divergence operators and odd Poisson brackets.Ann. Inst. Fourier, 52:419456, 2002.

[14] J.-L. Koszul. Crochet de Schouten-Nijenhuis et cohomologie.Asterisque, hors serie, pages257271, 1985.

[15] K. Mackenzie. Lie groupoids and Lie algebroids in Differential Geometry. Cambridge Uni-versity Press, 1987.

[16] E. Nelson.Tensor analysis. Princeton University Press, Princeton, 1967.

[17] P. Xu. Gerstenhaber algebras and BV-algebras in Poisson geometry. Comm. Math. Phys.,200:545560, 1999.


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HOMOLOGY AN D MODULAR CLAS SES OF LIE ALGEBROIDS 11

[18] A. Weinstein. The modular automorphism group of a Poisson manifold. J. Geom. Phys.,23:379394, 1997.

[19] E. Witten. Supersymmetry and Morse theory. J. Diff. Geom., 17:661692, 1982.

J. Grabowski: Mathematical Institute, Polish Academy of Sciences, Sniadeckich 8,

P.O. Box 21, 00-956 Warszawa, Poland

E-mail address: [email protected]

G. Marmo: Dipartimento di Scienze Fisice, Universita di Napoli Federico II and INFN,

Sezione di Napoli, via Cintia, 80126 Napoli, Italy


P. Michor: Institut fur Mathematik, Universitat Wien, Nordbergstrasse 15, A-1090

Wien, Austria; and: Erwin Schrodinger Institut fur Mathematische Physik, Boltzman-

ngasse 9, A-1090 Wien, Austria


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