E-Courant algebroids

0.35 MB·

by  Zhuo Chen

E-Courant Algebroids ∗
Zhuo Chen1, Zhangju Liu2 and Yunhe Sheng3
1Department of Mathematics,
Tsinghua University, Beijing 100084, China
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1 2Department of Mathematics and LMAM
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Peking University, Beijing 100871, China
2
3Department of Mathematics
r
a Jilin University, Changchun 130012, Jilin, China
M
email: [email protected], [email protected], [email protected]

3
2

] Abstract
G
Inthispaper,weintroducethenotionofE-Courantalgebroids,whereEisavectorbundle.
D
ItisakindofgeneralizedCourantalgebroidandcontainsCourantalgebroids,Courant-Jacobi
.
h algebroidsandomni-Liealgebroidsasitsspecialcases. Weexplorenovelphenomenaexhibited
t byE-Courantalgebroids andprovidemanyexamples. Westudytheautomorphism groupsof
a
omni-Lie algebroids and classify the isomorphism classes of exact E-Courant algebroids. In
m
addition, we introducethe concepts of E-Liebialgebroids and Manin triples.
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2
Contents
v
3
9 1 Introduction 1
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4 2 E-Courant algebroids 2
.
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0 3 The E-dual pair of Lie algebroids 9
8
0 4 The automorphism groups of omni-Lie algebroids 14
:
v
5 Exact E-Courant algebroids 17
i
X
r 6 E-Lie bialgebroids 21
a
7 Manin Triples 24
1 Introduction
In recent years, Courant algebroids are widely studied from several aspects. They are applied
in many mathematical objects such as Manin pairs and moment maps [1, 3, 16, 21], generalized
0Keyword:E-Courantalgebroids,E-Liebialgebroids,omni-Liealgebroids,Leibnizcohomology.
0MSC:Primary17B65. Secondary18B40,58H05.
∗Research partially supported by NSFC grants 10871007 and 10911120391/ A0109. The third author is also
supportedbyCPSFgrant20090451267.
1
complex structures [2, 11, 36], L -algebras and symplectic supermanifolds [31], gerbes [34], BV
∞
algebras and topological field theories [14, 32].
WerecalltwonotionscloselyrelatedtoCourantalgebroids—Jacobibialgebroidsandomni-Lie
algebroids. JacobibialgebroidsandgeneralizedLie bialgebroidsareintroduced, respectively,in [9]
and [15] to generalize Dirac structures from Poisson manifolds to Jacobi manifolds. More general
geometricobjectsaregeneralizedCourantalgebroids[29]andCourant-Jacobialgebroids[10]. The
notionofomni-Liealgebroids,ageneralizationofthenotionofomni-Liealgebrasintroducedin[39],
is defined in [5] in order to characterizeall possible Lie algebroidstructures on a vectorbundle E.
Anomni-LiealgebracanberegardedasthelinearizationoftheexactCourantalgebroidTM⊕T∗M
at a point and is studied from several aspects recently [2, 17, 35, 38]. Moreover, Dirac structures
of omni-Lie algebroids are studied by the authors in [6].
Inthispaper,weintroduceakindofgeneralizedCourantalgebroidcalledE-Courantalgebroids.
ThevaluesoftheanchormapofanE-CourantalgebroidlieinDE,thebundle ofdifferentialoper-
ators. Moreover,its Diracstructures arenecessarilyLie algebroidsequipped witha representation
on E. The notion of E-Courant algebroids not only unifies Courant-Jacobi algebroids and omni-
Lie algebroids,but also provides a number of interesting objects, e.g. the T∗M-Courantalgebroid
structure on the jet bundle of a Courant algebroid over M (Theorem 2.13).
Recall that an exact Courant algebroid structure on TM ⊕T∗M is a twist of the standard
Courant algebroid by a closed 3-form [34]. This structure includes twisted Poisson structures and
is related to gerbes and topological sigma models [30, 37]. In this paper, we are inspired to study
exact E-Courant algebroids similar to the situation of exact Courant algebroids.
Wealsostudytheautomorphismgroupsofomni-Liealgebroids,forwhichweneedthelanguage
of Leibniz cohomologies [25, 26]. Moreover, we introduce the notion of E-Lie bialgebroids, which
generalizesthenotionofgeneralizedLiebialgebroids. Weshallprovethat,foranE-Liebialgebroid,
there induces on the underlying vector bundle E a Lie algebroid structure (rank(E) ≥ 2), or a
local Lie algebra structure (rank(E)=1) (Theorem 6.6).
Thispaperisorganizedasfollows. InSection2weintroducethenotionofE-Courantalgebroids.
WeprovethatthejetbundleJC ofaCourantalgebroidC overM admitsanaturalT∗M-Courant
algebroid structure. In Section 3 we discuss the properties of E-dual pairs of Lie algebroids. In
Section 4 we find the automorphism groups and all possible twists of omni-Lie algebroids. In
Section 5 we study exact E-Courant algebroids and prove that every exact E-Courant algebroid
with an isotropicsplitting is isomorphicto anomni-Lie algebroid. In general,an exactE-Courant
algebroid is a twist of the standard omni-Lie algebroid by a 2-cocycle in the Leibniz cohomology
of Γ(DE) with coefficients in Γ(JE), which can also be treated as a 3-cocycle in the Leibniz
cohomology of Γ(DE) with coefficients in Γ(E). In Section 6 we study E-Lie bialgebroids. In
Section 7 we extend the theory of Manin triples from the context of Lie bialgebroids to E-Lie
bialgebroids and give some interesting examples.
Acknowledgement: Z. Chen would like to give his warmest thanks to P. Xu and M. Grutz-
mannfortheirusefulcomments. Y.ShenggiveshiswarmestthankstoL.Hoevenaars,M.Crainic,
I.MoerdijkandC.ZhufortheirusefulcommentsduringhisstayinUtrechtUniversityandCourant
Research Center, G¨ottingen. We also give our warmest thanks to the referees for many helpful
suggestions and pointing out typos and erroneous statements.
2 E-Courant algebroids
Let E→M be a vector bundle and DE the associated covariant differential operator bundle.
Known as the gauge Lie algebroid of the frame bundle F(E) (see [27, Example 3.3.4]), DE is
a transitive Lie algebroid with the Lie bracket [·,·] (commutator). The corresponding Atiyah
D
2
sequence is as follows:
0 //gl(E) i //DE j //TM //0. (1)
In [5], the authors proved that the jet bundle JE (see [7, 33] for more details about jet bundles)
can be regarded as an E-dual bundle of DE, i.e.
JE ∼= {ν ∈Hom(DE,E)|ν(Φ)=Φ◦ν(1E), ∀ Φ∈gl(E)}⊂Hom(DE,E).
Associated to the jet bundle JE, the jet sequence of E is given by:
0 //Hom(TM,E) e //JE p //E //0. (2)
The operator :Γ(E)→Γ(JE) is given by:
d
u(d):=d(u), ∀ u∈Γ(E), d∈Γ(DE).
d
The following formula is needed.
(fX)=df ⊗X +f X, ∀ X ∈Γ(C), f ∈C∞(M). (3)
d d
For a vector bundle K over M and a bundle map ρ:K−→DE, we denote the induced E-adjoint
bundle map by ρ⋆, i.e.
ρ⋆ :Hom(DE,E)→Hom(K,E), ρ⋆(ν)(k)=ν(ρ(k)), ∀ k ∈K, ν ∈Hom(DE,E). (4)
The notion of Leibniz algebras is introduced by Loday [24, 25, 12]. A Leibniz algebra g is an
R-module, where R is a commutative ring, endowedwith a linear map [·,·]:g⊗g−→g satisfying
[g ,[g ,g ]]=[[g ,g ],g ]+[g ,[g ,g ]], ∀ g ,g ,g ∈g.
1 2 3 1 2 3 2 1 3 1 2 3
Definition 2.1. An E-Courant algebroid is a quadruple (K,(·,·) ,[·,·] ,ρ), where
E K
• K is a vector bundle over M such that (Γ(K),[·,·] ) is a Leibniz algebra;
K
• (·,·) : K ⊗ K → E is a symmetric nondegenerate E-valued pairing, which induces an
E
embedding: K֒→Hom(K,E);
• the anchor ρ: K → DE is a bundle map,
such that the following properties hold for all X,Y,Z ∈Γ(K):
(EC-1) ρ[X,Y] =[ρ(X),ρ(Y)] ;
K D
(EC-2) [X,X] =ρ⋆ (X,X) ;
K d E
(EC-3) ρ(X)(Y,Z) =([X,Y] ,Z) +(Y,[X,Z] ) ;
E K E K E
(EC-4) ρ⋆(JE)⊂K, i.e. (ρ⋆(µ), X) = 1µ(ρ(X)), ∀ µ∈JE;
E 2
(EC-5) ρ◦ρ⋆ =0.
Remark 2.2. If the E-valued pairing (·,·) : K⊗K −→ E is surjective, Properties (EC-4) and
E
(EC-5) can be inferred from Property (EC-2). In particular, if E is a line bundle, any nondegen-
erate E-valued pairing (·,·) is surjective.
E
3
Lemma 2.3. For any X,Y ∈Γ(K) and f ∈C∞(M), we have
[X,fY] = f[X,Y] +( ◦ρ(X)f)Y, (5)
K K
j
[fX,Y] = f[X,Y] −( ◦ρ(Y)f)X +2ρ⋆(df ⊗(X,Y) ). (6)
K K j E
Proof. By Property (EC-3), for all X, Y, Z ∈Γ(K) and f ∈C∞(M), we have
([X,fY] ,Z) +(fY,[X,Z] ) = ρ(X)(fY,Z)
K E K E E
= ◦ρ(X)(f)(Y,Z) +fρ(X)(Y,Z)
j E E
= ◦ρ(X)(f)(Y,Z) +f([X,Y] ,Z) +f(Y,[X,Z] ) .
j E K E K E
Since the pairing (·,·) is nondegenerate, it follows that
E
[X,fY] = ◦ρ(X)(f)Y +f[X,Y] .
K K
j
By Property (EC-2), we have
[X,fY] +[fY,X] =2ρ⋆ (f(X,Y) )=2fρ⋆ (X,Y) +2ρ⋆(df ⊗(X,Y) ).
K K d E d E E
Substitute [X,fY] by (5) and apply Property (EC-2) again, we obtain (6).
K
For a subbundle L⊂K, denote by L⊥ ⊂K the subbundle
L⊥ ={e∈K| (e,l) =0, ∀ l∈L}.
E
Definition 2.4. A Dirac structure of an E-Courant algebroid (K,(·,·) ,[·,·] ,ρ) is a subbundle
E K
L⊂K which is closed under the bracket [·,·] and satisfies L=L⊥.
K
Evidently, L = L⊥ implies that L is maximal isotropic with respect to the E-valued pairing
(·,·) . In general, L being maximal isotropic with respect to (·,·) does not imply L=L⊥.
E E
Example 2.5. Let K be R3 with thestandard basis e1, e2, e3. The R-valued pairing (·,·)R is given
by
(e1,e3)R =(e2,e2)R =1, (e1,e1)R =(e1,e2)R =(e2,e3)R =(e3,e3)R =0.
Obviously, L=Re is maximal isotropic but L⊥ =Re ⊕Re 6=L.
1 1 2
Proposition 2.6. Any Dirac structure L has an induced Lie algebroid structure and is equipped
with a Lie algebroid representation ρ =ρ| : L → DE on E.
L L
Proof. Given a Dirac structure L, by Property (EC-2), we have [X,X] = 0, for all X ∈ Γ(L),
K
which implies that [·,·] | is skew-symmetric. By (5), (L,[·,·] | ,( ◦ρ) | ) is a Lie algebroid.
K L K L L
j
Finally by Property (EC-1), ρ : L → DE is a representation.
L
Remark 2.7. If E is the trivial line bundle M ×R, then DE ∼= TM ⊕(M ×R). Thus we can
decompose ρ=a+θ, for some a:K−→TM and θ :K−→M×R. For a Dirac structureL, since
ρ is a representation of the Lie algebroid L, it follows that θ =θ| ∈Γ(L∗) is a 1-cocycle in the
L L L
Lie algebroid cohomology of L. Therefore, (L,θ ) is a Jacobi algebroid, which is, by definition, a
L
Lie algebroid A together with a 1-cocycle θ ∈Γ(A∗) in the Lie algebroid cohomology [10].
One may refer to [27] for more general theories of Lie algebroids, Lie algebroid cohomologies
and their representations. Now we briefly recall the notions of omni-Lie algebroids, generalized
Courant algebroids, Courant-Jacobialgebroids and generalized Lie bialgebroids. We will see that
E-Courant algebroids unify all these structures.
4
• Omni-Lie algebroids
The notion of omni-Lie algebroids is introduced in [5] to characterize Lie algebroid structures
on a vector bundle. It is a generalization of Weinstein’s omni-Lie algebras. Recall that there is a
natural symmetric nondegenerate E-valued pairing h·,·i between JE and DE:
E
hµ,di =hd,µi , du, ∀ µ=[u] ∈JE, u∈Γ(E), d∈DE.
E E m
Moreover,this pairing is C∞(M)-linear and satisfies the following properties:
hµ,Φi = Φ◦ (µ), ∀ Φ∈gl(E), µ∈JE;
E p
hy,di = y◦ (d), ∀ y∈Hom(TM,E), d∈DE.
E j
Furthermore, Γ(JE) is invariant under any Lie derivative L , d∈Γ(DE), which is defined by the
d
Leibniz rule:
hL µ,d′i ,dhµ,d′i −hµ,[d,d′] i , ∀ µ∈Γ(JE), d′ ∈Γ(DE). (7)
d E E D E
Definition 2.8. [5] Given a vector bundle E, the quadruple (E,{·,·},(·,·) ,ρ) is called the omni-
E
Lie algebroid associated to E, where E =DE⊕JE, the anchor ρ is the projection from E to DE,
the bracket operation {·,·} and the nondegenerate E-valued pairing (·,·) are given respectively by
E
1
(d+µ,r+ν) , (hd,νi +hr,µi ), (8)
E 2 E E
{d+µ,r+ν} , [d,r] +L ν−L µ+ hµ,ri . (9)
D d r d E
If there is no risk of confusion, we simply denote the omni-Lie algebroid (E,{·,·},(·,·) ,ρ) by
E
E. WecalltheE-valuedpairing(8)andthebracket(9),respectively, the standard pairing and
the standard bracket onE =DE⊕JE. One mayreferto [5] formore details ofthe propertyof
omni-Lie algebroids. Evidently, the E-adjoint map ρ⋆ is 1 , the identity map on JE. It is easily
JE
seen that the omni-Lie algebroidE is an E-Courant algebroid. Its Dirac structures are studied by
the authors in [6].
• Generalized Courant algebroids (Courant-Jacobi algebroids)
ThenotionofgeneralizedCourantalgebroidsisintroducedin[29]. Itisapair(K,ρθ)subjectto
some compatibility conditions, where K → M is a vector bundle equipped with a nondegenerate
symmetric bilinear form (·,·), a skew-symmetricbracket[·,·] on Γ(K) anda bundle map ρθ :K →
TM ×R, which is also a first-order differential operator. We may write ρθ(X) = (ρ(X),hθ,Xi),
where ρ: K → TM is linear and θ ∈Γ(K∗) satisfies
θ([X,Y])=ρ(X)θ(Y)−ρ(Y)θ(X), ∀ X,Y ∈Γ(K).
One should note that the skew-symmetric bracket [·,·] does not satisfy the Jacobi identity. The
notion of Courant-Jacobialgebroids is introduced in [10]. In [29], it is established the equivalence
ofgeneralizedCourantalgebroidsandCourant-Jacobialgebroids. Roughlyspeaking,thedifference
between them is that the generalized Courant algebroid has a skew-symmetric bracket [·,·] and a
Courant-Jacobialgebroidhasanoperation◦,whichisalsoknownastheDorfmanbracket[8]. The
former does not satisfy the Jacobi identity, while the later satisfies the Leibniz rule. Moreover,
[·,·] can be realized as the skew-symmetrization of ◦. A generalized Courant algebroid reduces to
a Courant algebroid if θ =0 (see [22]).
Evidently, a generalized Courant algebroid is an E-Courant algebroid if we take E = M ×R.
It follows that all Jacobi algebroids and Courant algebroids are M ×R-Courant algebroids.
5
• Generalized Lie bialgebroids
A Lie bialgebroid is a pair of vector bundles in duality, each of which is a Lie algebroid, such
thatthe differentialdefined byone ofthem onthe exterioralgebraofits dualis a derivativeofthe
Schouten bracket [18, 28]. A generalized Lie bialgebroid [15], or a Jacobi bialgebroid [9], is a pair
((A,φ ),(A∗,X )), where A and A∗ are two vector bundles in duality, and, respectively, equipped
0 0
with Lie algebroid structures (A,[·,·],a) and (A∗,[·,·] ,a ). The data φ ∈Γ(A∗) and X ∈Γ(A)
∗ ∗ 0 0
are 1-cocycles in their respective Lie algebroid cohomologies such that for all X,Y ∈ Γ(A), the
following conditions are satisfied:
d [X,Y]=[d X,Y] +[X,d Y] , (10)
∗X0 ∗X0 φ0 ∗X0 φ0
φ (X )=0, a(X )=−a (φ ), L X +L X =0, (11)
0 0 0 ∗ 0 ∗φ0 X0
where d is the X -differential of A, [·,·] is the φ -Schouten bracket, L and L are the usual
∗X0 0 φ0 0 ∗
Liederivatives. Formoreinformationofthesenotations,pleasereferto[15]. ForaJacobimanifold
(M,X,Λ), ((TM ×R,(0,1)),(T∗M ×R,(−X,0))) is a generalized Lie bialgebroid. Furthermore,
for a generalizedLie bialgebroid,there is an induced Jacobistructure on the base manifold M. In
particular, both ((A,φ ) and (A∗,X )) are Jacobi algebroids. If φ = 0 and X =0, a generalized
0 0 0 0
Lie bialgebroid reduces to a Lie bialgebroid. It is known that for a generalized Lie bialgebroid
((A,φ ),(A∗,X )), there is a natural generalized Courant algebroid (A⊕A∗,φ +X ).
0 0 0 0
We give more examples of E-Courant algebroids.
Example 2.9. Let A be a Lie algebroid and ρ : A → DE a representation of A on a vector
A
bundle E. Let K=A⊕(A∗⊗E). For any X, Y ∈Γ(A), ξ⊗u, η⊗v ∈Γ(A∗⊗E), we define the
following operations:
ρ(X +ξ⊗u) = ρ (X),
A
[X +ξ⊗u,Y +η⊗v] = [X,Y]+L (η⊗v)−L (ξ⊗u)+ρ⋆ ◦ (hY,ξiu),
K X Y A d
1
(X +ξ⊗u,Y +η⊗v) = (hX,ηiv+hY,ξiu).
E 2
Evidently, ρ⋆ = ρ⋆ : JE −→ A∗ ⊗ E and it is straightforward to check that (A ⊕ (A∗ ⊗
A
E),[·,·] ,(·,·) ,ρ) is an E-Courant algebroid. In [20], the notion of AV-Courant algebroids is
K E
introduced in order to study generalized CR structures, which is closely related to this example
but twisted by a 3-cocycle in the cohomology of the Lie algebroid representation ρ .
A
Example 2.10. Consider an E-Courant algebroid K whose anchor ρ is zero. Thus ρ⋆ = 0, and
thebracket[·,·] isskew-symmetric. SoKisabundleofLiealgebras. Property(EC-3)showsthat
K
there is an invariantE-valued pairing. We conclude that an E-CourantalgebroidK whose anchor
ρ is zero is equivalently a bundle of Lie algebras with an invariant E-valued pairing.
Example2.11. Anomni-Liealgebragl(V)⊕V isaspecialomni-Liealgebroidwhosebasemanifold
is a point, hence a V-Courant algebroid. Moreover, one may consider a Lie algebra (g,[·,·] )
g
with faithful representation ρ : g −→ gl(V) on a vector space V. This representation is called
g
nondegenerate if for any v ∈ V, there is some A ∈ g such that ρ (A)(v) 6= 0. Introduce a
g
nondegenerate V-valued pairing (·,·) and a bilinear bracket [·,·] on the space g⊕V:
V
1
(A+u,B+v) = (ρ (A)(v)+ρ (B)(u)),
V g g
2
[A+u,B+v] = [A,B] +ρ (A)(v), ∀ A+u, B+v ∈g⊕V,
g g
where ρ:g⊕V −→gl(V) is defined by ρ(A+u)=ρ (A) for A+u∈g⊕V. Following from
g
1
ρ⋆(u)(B+v)= ρ (B)(u)=(u,B) , (12)
g V
2
6
wehaveρ⋆ =1 ,asamapJV =V −→V. Clearly,(g⊕V,(·,·) ,[·,·],ρ)isaV-Courantalgebroid.
V V
The bracket defined above appeared in [17]. For any representation ρ : g −→ gl(V), we call
(g⊕V,[·,·])ahemisemidirectproductofgwithV. ThereisalsoanaturalexactCourantalgebra
associated to any g-module [2].
The above example can be generalized to the situation of Lie algebroids.
Example 2.12. Let(A,[·,·],a)beaLiealgebroidwithanondegeneraterepresentationρ :A−→
A
DE. On the vector bundle A⊕JE, define an E-valued pairing (·,·) and a bracket {·,·} by
E
1
(X +µ,Y +ν) = (hρ (X),νi +hρ (Y),µi ),
E 2 A E A E
{X+µ,Y +ν} = [X,Y]+L ν−L µ+ hρ (Y),µi ,
ρ(X) ρ(Y) d A E
for any X+µ, Y +ν ∈Γ(A⊕JE), and define ρ:A⊕JE −→DE by ρ(X+µ)=ρ (X). Similar
A
to (12), we have ρ⋆ = 1 . Then, it is easily seen that (A⊕JE,(·,·) ,{·,·},ρ) is an E-Courant
JE E
algebroid.
• The jet bundle of a Courant algebroid
At the end of this section, we prove that for any Courant algebroid C, JC is a T∗M-Courant
algebroid. The original definition of a Courant algebroid is introduced in [22]. Here we use the
alternative definition given by D. Roytenberg in [31], that a Courant algebroid is a vector bundle
C −→M together with some compatible structures — a nondegenerate bilinear form h·,·i on the
bundle, a bilinear operation J·,·K on Γ(E) and a bundle map a: C −→TM satisfying a◦a∗ =0.
In particular, (Γ(C),J·,·K) is a Leibniz algebra.
On the jet bundle JC of the vector bundle C, we introduce the T∗M-valued pairing (·,·) , the
∗
bracket [·,·] and the anchor ρ:JC −→D(T∗M) as follows.
JC
a) For any X, Y ∈Γ(C), the T∗M-valued pairing (·,·) of X, Y is given by
∗ d d
( X, Y) =dhX,Yi. (13)
d d ∗
By (3), we get
( X,df ⊗Y) = hX,Yidf,
d ∗
(df ⊗X,df ⊗Y) = 0.
∗
b) For any X, Y ∈Γ(C), the bracket [·,·] of X, Y is given by
JC
d d
[ X, Y] = JX,YK. (14)
JC
d d d
By (5), (6) and (3), we have
[ X,df ⊗Y] = df ⊗JX,YK+d(a(X)f)⊗Y,
JC
d
[df ⊗Y, X] = df ⊗JY,XK−d(a(X)f)⊗Y +2hX,Yi a∗(df),
JC
d d
[df ⊗X,dg⊗Y] = a(X)(g)df ⊗Y −a(Y)(f)dg⊗X.
JC
c) For any X ∈Γ(C), ρ( X)∈Γ(D(T∗M)) is given by
d
ρ( X)(·)=L (·). (15)
a(X)
d
By (3), we get
ρ(df ⊗X)=a(X)⊗df, ∀ f ∈C∞(M).
7
For any ξ ∈Ω1(M), we have
ρ( X)(fξ)=L (fξ)=fL (ξ)+a(X)(f)ξ,
a(X) a(X)
d
which implies that ◦ρ◦ X =a(X), where :D(T∗M)−→TM is the anchor of D(T∗M)
j d j
givenin(1). Furthermore,foranyg ∈C∞(M),thefactthatρ(df⊗X)(gξ)=gρ(df⊗X)(ξ)
implies that ◦ρ(df ⊗X)=0.
j
We identify C with C∗ by the bilinear form. For any f, g ∈ C∞(M), it is straightforward to
obtain the following relations:
ρ⋆( df)= (a∗df),
ρ⋆(df ⊗ddg)=ddg⊗a∗(df), (16)

ρ⋆( (fdg))=dg⊗a∗(df)+f (a∗dg).

d d
These structures give rise to a T∗M-Courantalgebroid.

Theorem 2.13. For any Courant algebroid C, (JC,(·,·) ,[·,·] ,ρ) is a T∗M-Courant algebroid.
∗ JC
Proof.Itisstraightforwardtoseethatthepairing(·,·) andρarebundlemapsand(Γ(JC),[·,·] )
∗ JC
is a Leibniz algebra. To show that the data (JC,(·,·) ,[·,·] ,ρ) satisfies the properties listed in
∗ JC
Definition 2.1, it suffices to consider elements of the form X, Y, Z, df ⊗X, dg⊗Y, dh⊗Z,
d d d
where X, Y, Z ∈Γ(C), f, g, h∈C∞(M).
First we check Property (EC-1). Clearly, we have
ρ[ X, Y] =ρ JX,YK=L =[L ,L ] =[ρ X,ρ Y] .
JC aJX,YK a(X) a(Y) D D
d d d d d
Furthermore, since a◦a∗ =0, we have
ρ[df ⊗X, Y] = ρ(df ⊗JX,YK−d(a(Y)f)⊗X +2hX,Yi a∗(df))
JC
d d
= a([X,Y])⊗df −a(X)⊗d(a(Y)f).
On the other hand,
[ρ(df ⊗X),ρ( Y)] (ξ) = [a(X)⊗df,L ] (ξ)=hL ξ,a(X)idf −L (ha(X),ξidf)
D a(Y) D a(Y) a(Y)
d
= ha([X,Y]),ξidf −ha(X),ξid(a(Y)f),
which implies
ρ[df ⊗X, Y] =[ρ(df ⊗X),ρ( Y)] .
JC D
d d
Similarly, we have
ρ[ X,df ⊗Y] = [ρ( X),ρ(df ⊗Y)]
JC D
d d
= a([X,Y])⊗df +a(Y)⊗d(a(X)f),
ρ[df ⊗X,dg⊗Y] = [ρ(df ⊗X),ρ(dg⊗Y)]
JC D
= (a(X)g)a(Y)⊗df −(a(Y)f)a(X)⊗dg.
To see Property (EC-2), notice that [df ⊗X,df ⊗X] = 0 and (df ⊗X,df ⊗X) = 0, so we
JC ∗
have
[df ⊗X,df ⊗X] =ρ⋆ (df ⊗X,df ⊗X) .
JC d ∗
Furthermore,
[ X, X] = JX,XK= ◦a∗(dhX,Xi)=ρ⋆ ◦dhX,Xi=ρ⋆ ( X, X) ,
d d JC d d d d d d ∗
[df ⊗X, Y] +[ Y,df ⊗X] = 2hX,Yiρ⋆ df +2df ⊗a∗(dhX,Yi)
JC JC
d d d
= 2ρ⋆ (hX,Yidf)=2ρ⋆ (df ⊗X,Y) ,
d d ∗
whichimpliesthatProperty(EC-2)holds. ItisstraightforwardtoverifyProperty(EC-3). Property
(EC-4) follows from (16). Property (EC-5) follows from the fact that a◦a∗ =0.
8
3 The E-dual pair of Lie algebroids
Let A be a vector bundle and B a subbundle of Hom(A,E). For any µk ∈ Hom(∧kA,E), denote
by µk the induced bundle map from ∧k−1A to Hom(A,E) such that
♮
µk(X ,··· ,X )(X )=µk(X ,··· ,X ,X ). (17)
♮ 1 k−1 k 1 k−1 k
Introduce a series of vector bundles Hom(∧kA,E) , k ≥ 0 by setting Hom(∧0A,E) = E,
B B
Hom(∧1A,E) =B and
B
Hom(∧kA,E) , µk ∈Hom(∧kA,E) | Im(µk)⊂B , (k ≥2). (18)
B ♮
IfBisasubbundleofHom(A,E),the(cid:8)nAisalsoabundleofHom(B,E).(cid:9)ThenotationHom(∧kB,E)
A
is thus clear.
Definition 3.1. Let A and E be two vector bundles over M. A vector bundle B ⊂Hom(A,E) is
called an E-dual bundle of A if the E-valued pairing h·,·i : A× B → E, ha,bi ,b(a) (where
E M E
a∈A, b∈B) is nondegenerate.
Obviously, if B is an E-dual bundle of A, then A is an E-dual bundle of B. We call the pair
(A,B) an E-dual pair of vector bundles.
Assume that (A,[·,·],a) is a Lie algebroid and B ⊂ Hom(A,E) is an E-dual bundle of A. A
representation ρ : A → DE of A on E is said to be B-invariant if (Γ(Hom(∧•A,E) ),dA) is
A B
a subcomplex of (Ω•(A,E),dA), where dA is the coboundary operatorassociatedto ρ . If ρ is a
A A
B-invariant representation, we have ρ⋆(JE)⊂B. In fact, by definition, one has
A
ρ⋆(µ)(X)=hµ,ρ (X)i , ∀ µ∈JE, X ∈A,
A A E
and it follows that ρ⋆ : JE → B is given by ρ⋆([u] ) = (dAu) , for all u ∈ Γ(E). Thus,
A A m m
ρ⋆(JE)⊂B is equivalent to the condition that dA(Γ(E))⊂Γ(B).
A
Furthermore, for any representation ρ : A → DE, there are two natural Lie derivative
A
operations along X ∈Γ(A). The first one is
L :Γ(Hom(∧kA,E))−→Γ(Hom(∧kA,E))=Γ(∧kA∗⊗E)
X
defined by
L (ω⊗u)=(L ω)⊗u+ω⊗ρ (X)u, ∀ ω ∈Γ(∧kA∗), u∈Γ(E).
X X A
The second one is
L :Γ(Hom(∧k(A∗⊗E),E))−→Γ(Hom(∧k(A∗⊗E),E))=Γ(∧k(A⊗E∗)⊗E)
X
defined by L u=ρ (X)u, for u∈Γ(E), and
X A
k
L Ξ(̟ ∧···∧̟ )=ρ (X)(Ξ(̟ ∧···∧̟ ))− Ξ(̟ ∧···∧L ̟ ∧···∧̟ ), (19)
X 1 k A 1 k 1 X i k
i=1
X
for all Ξ ∈ Γ(Hom(∧k(A∗⊗E),E)), ̟ ∈ Γ(A∗ ⊗E). In particular, since A ⊂ Hom(A∗ ⊗E,E),
i
we have
L Y =[X,Y], ∀ Y ∈Γ(A).
X
9
Proposition 3.2. Let A be a Lie algebroid together with a representation ρ : A −→ DE and
B ⊂ Hom(A,E) a subbundle of Hom(A,E) such that (A,B) is an E-dual pair of vector bundles.
Then the following statements are equivalent:
(1) the representation ρ : A → DE is B-invariant;
A
(2) dAΓ(E)⊂Γ(B) and dAΓ(B)⊂Γ(Hom(∧2A,E) );
B
(3) Γ(Hom(∧kA,E) ) is invariant under the operation L for any X ∈Γ(A);
B X
(4) Γ(Hom(∧kB,E) ) is invariant under the operation L for any X ∈Γ(A).
A X
Proof.Theimplication(1)=⇒(2)isobvious. Weadoptaninductiveapproachtoseetheimplica-
tion(2)=⇒(1). Foranyn≥1,L :Γ(Hom(∧nA,E) )−→Γ(Hom(∧nA,E) )iswelldefinedand
X B B
we have i L −L i = i . Assume that dAΓ(Hom(∧n−1A,E) ) ⊂ Γ(Hom(∧nA,E) ) and
Y X X Y [Y,X] B B
dAΓ(Hom(∧nA,E) )⊂ Γ(Hom(∧n+1A,E) ) hold for all µn+1 ∈Γ(Hom(∧n+1A,E) ). To prove
B B B
that dAµn+1 ∈ Γ(Hom(∧n+2A,E) ), it suffices to show that i dAµn+1 ∈ Γ(Hom(∧n+1A,E) ),
B X B
for all a ∈ Γ(A). Again, it suffices to show that i i dAµn+1 ∈ Γ(Hom(∧nA,E) ) holds for all
Y X B
Y ∈Γ(A). In fact,
i i dAµn+1 = i (L µn+1−dAi µn+1)
Y X Y X X
= (i L −L i )µn+1+L i µn+1−i dAi µn+1
Y X X Y X Y Y X
= i µn+1+L i µn+1−i dAi µn+1 ∈Γ(Hom(∧nA,E) ).
[Y,X] X Y Y X B
So we conclude that Γ(Hom(∧•A,E) ) is a subcomplex of Ω•(A,E). This completes the proof of
B
the equivalence of (1) and (2). The equivalence of (1) and (3) is obvious.
Nextwe provethe equivalence of(2)and(4). For anyXk ∈Γ(Hom(∧kB,E) )andξ ∈B, we
A i
have
i L Xk,ξ
ξ1∧···∧ξk−1 X k E
= ((cid:10)LXXk)(ξ1∧ξ2∧···(cid:11)∧ξk)
k
= ρ (X)(Xk(ξ ∧ξ ∧···∧ξ ))− Xk(ξ ∧···∧L ξ ∧···∧ξ )
A 1 2 k 1 X i k
i=1
X
k−1
= ρ (X) i Xk,ξ − i Xk,ξ − i Xk,L ξ
A ξ1∧···∧ξk−1 k E ξ1∧···∧LXξj∧···∧ξk−1 k E ξ1∧···∧ξk−1 X k E
j=1
(cid:10) (cid:11) X(cid:10) (cid:11) (cid:10) (cid:11)
k−1
= [X,i Xk]− i Xk,ξ .
ξ1∧···∧ξk−1 ξ1∧···∧LXξj∧···∧ξk−1 k
* +
Xj=1 E
Since the E-valued pairing h·,·i is nondegenerate, we have
E
k−1
i L Xk =[X,i Xk]− i Xk,
ξ1∧···∧ξk−1 X ξ1∧···∧ξk−1 ξ1∧···∧LXξj∧···∧ξk−1
j=1
X
which implies the equivalence of (2) and (4).
Definition 3.3. An E-dual pair of Lie algebroids ((A,ρ );(B,ρ )) consists of two Lie alge-
A B
broids A and B which are mutuallyE-dual vector bundles, a B-invariant representation ρ : A →
A
DE and an A-invariant representation ρ : B → DE.
B
10