Dirac structures of omni-Lie algebroids

0.28 MB·English

by  Zhuo Chen

Dirac structures of omni-Lie algebroids ∗
Zhuo Chen1, Zhangju Liu2 and Yunhe Sheng3
1Department of Mathematics,
Tsinghua University, Beijing 100084, China
2Department of Mathematics and LMAM
9
0 Peking University, Beijing 100871, China
0 3Department of Mathematics
2
Jilin University, Changchun 130012, Jilin, China
l
u email: [email protected], [email protected], [email protected]
J

5

]
G Abstract
D Thegeneralized Courant algebroid structureattachedtothedirectsum E =DE⊕
JE for a vector bundle E is called an omni-Lie algebroid, as it is reduced to the
.
h omni-Lie algebra introduced by A. Weinstein if the base manifold is a point. A Dirac
at structureinE isnecessarily aLiealgebroid associated witharepresentationonE. We
m study the geometry underlyingthese Dirac structures in the light of reduction theory.
In particular, we prove that there is a one-to-one correspondence between reducible
[
Dirac structures of E and projective Lie algebroids in T =TM ⊕E; we establish the

2 relationbetweenthenormalizerNL ofareducibleDiracstructureLandthederivation
v algebraDer(b(L))oftheprojectiveLiealgebroidb(L);westudythecohomologygroup
9 H•(L,ρL) and the relation between NL and H1(L,ρL); we describe Lie bialgebroids
1
using the adjoint representation and the deformation of a Dirac structure, which is
8 related with H2(L,ρL).
3
.
2
Contents
0
8
0 1 Introduction 2
:
v
2 Omni-Lie Algebroids 3
i
X
3 Dirac Structures and Their Reductions 6
r
a
4 Some Examples 11
5 The Normalizer of Dirac Structures 14
6 Cohomology of Dirac Structures 17
0Keywords: gauge Lie algebroid, jet bundle, omni-Lie algebroid, Dirac structure, local Lie algebra, re-
duction,normalizer,deformation.
0MSC:17B66.
∗ResearchpartiallysupportedbyNSFC(10871007)andCPSF(20060400017); thethirdauthorfinancially
supportedbythegovernmental scholarshipfromChinaScholarshipCouncil.
1
1 Introduction
Lie algebroids(and local Lie algebrasin the sense of Kirillov [14]) are generalizationsof Lie
algebrasthatnaturallyappearinPoissongeometry(anditsvariations,e.g.,Jacobimanifolds
inthesenseofLichnerowicz[17])(see[21]foradetaileddescriptionofthissubject). Courant
algebroids are combinations of Lie algebroids and quadratic Lie algebras. It was originally
introduced in [8] by T. Courant where he first called them Dirac manifolds, and then were
re-namedafterhimin[20](seealsoanalternatedefinition[27])byLiu,WeinsteinandXuto
describethedoubleofaLiebialgebroid. Recently,severalapplicationsofCourantalgebroids
andDiracstructureshavebeenfoundindifferentfields,e.g.,Maninpairsandmomentmaps
[1],[4]; generalizedcomplex structures[3], [10]; L -algebrasandsymplectic supermanifolds
∞
[24]; gerbes [26] as well as BV algebras and topological field theories [12], [25].
Motivated by an integrability problem of the Courant bracket, A. Weinstein gives a
linearization of the Courant bracket at a point [31], which is studied from several aspects
recently ([3,13, 23, 28]). Since Dirac structures ofCourantalgebroidsarenaturalproviders
ofLiealgebroidsandA.Weinsteinhasshownthatanomni-Liealgebrastructurecanencode
allLiealgebrastructures,thenextstepis,logically,tofindoutcandidatesthatcouldencode
all Lie algebroid structures. In a recent work [6], we have given a definitive answer to this
question.
Let us first review the contents of [6]. A generalized Courant algebroid structure is
defined on the direct sum bundle DE⊕JE, where DE and JE are the gauge Lie algebroid
andthe jet bundle ofa vectorbundle E respectively. Sucha structureis calledanomni-Lie
algebroid since it reduces to the omni-Lie algebra introduced by A. Weinstein if the base
manifold is a point [31].
It is well known that the theory of Dirac structures has wide and deep applications in
both mathematics and physics (e.g., [2], [5], [9], [10], [11], [30]). In [6], only some special
Dirac structures were studied and it is proved that there is a one-to-one correspondence
between Dirac structures coming from bundle maps JE → DE and Lie algebroid (local
Lie algebra) structures on E when rank(E) ≥ 2 (E is a line bundle). In other words,
Dirac structures that are graphs of maps actually underlines the geometric objects of Lie
algebroids, or local Lie algebras.
Asacontinuationof[6],thepresentpaperexploreswhatageneralDiracstructureofthe
omni-Lie algebroidwould encode. As we shall see, for a vector space V, Dirac structures in
the omni-Lie algebra gl(V)⊕V come from Lie algebra structures on subspaces of V (this
coincides with Weinstein’s result [31]). For a vector bundle E over M, Dirac structures in
the omni-Lie algebroid E = DE⊕JE turn out to be more complicated than that of omni-
Lie algebras. The key concept we need is that of a projective Lie algebroid — a subbundle
A ⊂ T = TM ⊕E, which is equipped with a Lie algebroid structure such that the anchor
is the projection from A to TM. A Dirac structure L ⊂ E is called reducible if b(L) is a
regular subbundle of T. We shall see that any Dirac structure is reducible if rank(E) ≥ 2
(Lemma 3.1).
The main result is Theorem 3.7, which claims a one-to-one correspondence between
reducible Diracstructures inE andprojectiveLie algebroidsinT. Infact, the projectionof
a reducible Dirac structure L to T yields a projective Lie algebroidb(L) and, conversely, a
projective Lie algebroid A⊂T can be uniquely lifted to a Dirac structure LA by means of
a connection in E.
Furthermore,usingthefallingoperator(·) ,weestablishaconnectionbetweenthederiva-
•
2
tion algebra Der(A) of a projective Lie algebroid A and the normalizer N of the corre-
LA
sponding lifted Dirac structure LA. We prove that, for any X ∈ N , X ∈ Der(A).
LA •
Conversely, any δ ∈ Der(A) can be lifted to an element in N . Another observation is
LA
that, to any Dirac structure L ⊂ E, there associates a representation of L on E, namely
ρ : L −→ DE (Proposition 2.5). So there is an associated cohomology group H•(L,ρ ).
L L
We will see that the normalizer of L is related with H1(L,ρ ) and the deformation of L is
L
related with H2(L,ρ ).
L
Thispaperisorganizedasfollows. InSection2werecallthebasicpropertiesofomni-Lie
algebroids. InSection3,westatethemainresultofthispaper—thecorrespondencebetween
reducible Dirac structures and projective Lie algebroids. In Section 4, several interesting
examples are discussed. In Section 5, we study the relation between the normalizer of a
reducibleDiracstructureandLie derivations. InSection6,wegivesomeapplicationsofthe
related cohomologies of Dirac structures.
Acknowledgement: Z. Chen would like to thank P. Xu and M. Stienon for the useful
discussions and suggestions that helped him improving this work. Y.-H. Sheng gives his
warmest thanks to L. Hoevenaars, M. Crainic, I. Moerdijk and C. Zhu for their useful
commentsduringhisstayinUtrechtUniversity,whereapartofworkwasdoneandCourant
Research Center, G¨oettingen.
2 Omni-Lie Algebroids
We use the followingconventionthroughoutthe paper: E → M denotes a vectorbundle E
overasmoothmanifoldM (weassumethatE isnotazerobundle),d: Ω•(M) → Ω•+1(M)
the usualdeRhamdifferentialofforms andm anarbitrarypointinM. ByT we denote the
direct sum TM ⊕E and use pr , pr , respectively, to denote the projection from T to
TM E
TM and E.
First,webrieflyreviewthenotionofomni-Liealgebroidsdefinedin[6],whichgeneralizes
omni-Lie algebras defined by A. Weinstein in [31]. Given a vector bundle E, let JE be the
(1-)jet bundle of E ([22]), and DE the gauge Lie algebroid of E ([21]). These two vector
bundles associate, respectively, with the jet sequence:
// // // //
0 Hom(TM,E) e JE p E 0, (1)
and the Atiyah sequence:
0 // gl(E) i //DE α //TM // 0. (2)
Theembeddingmaps and intheabovetwoexactsequenceswillbeignoredwhenthereis
e i
noriskofconfusion. Itis wellknownthatDE is atransitiveLiealgebroidoverM,withthe
anchor α as above ([15]). The E-duality between two vector bundles is defined as follows.
Definition 2.1. Let A, B and E be vector bundles over M. We say that B is an E-dual
bundle of A if there is a C∞(M)-bilinear E-valued pairing h·,·i : A× B → E which
E M
is nondegenerate, that is, the map a 7→ ha,·i is an embedding of A into Hom(B,E), and
E
similarly for the B-entry.
Animportantresultin[6]isthatJEisanE-dualbundleofDEwithsomeniceproperties.
In fact, we have a nondegenerate E-pairing h·,·i between JE and DE:
E
hµ,di =hd,µi , du, ∀ µ=[u] ∈JE, u∈Γ(E), d∈DE.
E E m
3
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◦α(d), ∀ y∈Hom(TM,E), d∈DE.
E
An equivalent expression is that we can define JE by DE,
JE ∼= {ν ∈Hom(DE,E)|ν(Φ)=Φ◦ν(1E), ∀ Φ∈gl(E)}⊂Hom(DE,E).
Conversely, DE is also determined by JE:
DE ∼= {δ ∈Hom(JE,E)|∃ x∈TM, s.t. δ(y)=y(x), ∀ y∈Hom(TM,E)}.
For a Lie algebroid (A,[·,·],α) over M, a representation of A on a vector bundle E →
M is a Lie algebroid morphism L : A → DE. We may also refer to E as an A-
module. To such a representation, there associates a cochain complex Ωi(A,E) =
Pi≥0
Γ(Hom(∧iA,E)) with the coboundary operator:
Pi≥0
d :Ω•(A,E) → Ω•+1(A,E),
A
defined in a similar fashion as that of the deRham differential [21]. Since DE is a Lie
algebroid and E is a natural DE-module, we have the cochain complex:
Ω•(DE,E)=Γ(Hom(∧•DE,E))
with the coboundary operator:
:Ω•(DE,E) → Ω•+1(DE,E). (3)
d
Note that, ∀ u∈Γ(E), u∈Ω1(DE,E) is a section of JE and we have a formula:
d
(fu)=f u+df ⊗u, ∀ f ∈C∞(M), u∈Γ(E).
d d
ThesectionspaceΓ(JE)isaninvariantsubspaceoftheLiederivativeL foranyd∈Γ(DE).
d
Here L is defined by the Leibniz rule as follows:
d
hL µ,d′i , dhµ,d′i −hµ,[d,d′] i , ∀ µ∈Γ(JE), d′ ∈Γ(DE).
d E E D E
Definition 2.2. [6] We call the quadruple (E,{·,·},(·,·) ,ρ) an omni-Lie algebroid, where
E
E = DE⊕JE, ρ is the projection from E to DE, the bracket {·,·} : Γ(E)×Γ(E) −→ Γ(E)
is defined by
{d+µ,r+ν},[d,r] +L ν−L µ+ hµ,ri ,
D d r d E
and (·,·) is a nondegenerate symmetric E-valued 2-form on E defined by:
E
1
(d+µ,r+ν) , (hd,νi +hr,µi ),
E 2 E E
for any d, r∈DE, µ, ν ∈JE.
Theorem 2.3. [6] An omni-Lie algebroid satisfies the following properties, ∀ X, Y, Z ∈
Γ(E), f ∈C∞(M):
4
1) (Γ(E),{·,·}) is a Leibniz algebra,
2) ρ{X,Y}=[ρ(X),ρ(Y)] ,
D
3) {X,fY}=f{X,Y}+(α◦ρ(X))(f)Y,
4) {X,X}= (X,X) ,
d E
5) ρ(X)(Y,Z) =({X,Y},Z) +(Y,{X,Z}) .
E E E
From these, it is easy to obtain the following equalities:
{fX,Y}=f{X,Y}−(α◦ρ(Y))(f)Y +2df ⊗(X,Y) , (4)
E
{X,Y}+{Y,X}=2 (X,Y) . (5)
d E
For a subbundle S ⊂E, we denote
S⊥ ={X ∈E | (X,s) =0, ∀ s∈S}.
E
We call S isotropic with respect to (·,·) if S ⊂S⊥.
E
Definition 2.4. [6] A Dirac structure in the omni-Lie algebroid E is a maximal isotropic1
subbundle L⊂E such that {Γ(L),Γ(L)}⊂Γ(L).
Proposition 2.5. [6] A Dirac structure L is necessarily a Lie algebroid with the restricted
bracket and the anchor α◦ρ. Moreover, ρ =ρ| :L→DE is a representation of L on E.
L L
For T = TM ⊕E, we have the standard decomposition
Hom(T,E)=gl(E)⊕Hom(TM,E).
The following exact sequence will be referred as the omni-sequence of E.
// a // b // //
0 Hom(T,E) E T 0, (6)
where the maps a and b are defined, respectively, by
a(Φ+y)= (Φ)+ (y), ∀ Φ∈gl(E), y∈Hom(TM,E);
i e
b(d+µ)=α(d)+ (µ), ∀ d∈DE, µ∈JE.
p
WeregardHom(T,E)asasubbundleofE andomittheembeddinga. Evidently,Hom(T,E)
is a maximal isotropic subbundle of E. In fact, it is a Dirac structure of E and the bracket
is given by
{α,β}=α◦β−β◦α, ∀ α,β ∈Γ(Hom(T,E)).
In particular, if α = Φ+φ, β = Ψ+ψ, where Φ,Ψ ∈ Γ(gl(E)), φ,ψ ∈ Γ(Hom(TM,E)),
then
{Φ,Ψ}=Φ◦Ψ−Ψ◦Φ, {φ,ψ}=0, {Φ,φ}=Φ◦φ.
Lemma 2.6. (1) The subspace Γ(Hom(T,E)) is a right ideal of Γ(E).
1OnemayprovethatLismaximalisotropicifandonlyifL=L⊥.
5
(2) For any h∈Γ(Hom(T,E)), X ∈Γ(E), we have
b{h,X}=h(b(X)). (7)
Notethat (2)implies thatthe bracketofΓ(Hom(T,E)) andΓ(E) is fiber-wiselydefined.
Proof. For any X =d+µ∈Γ(E) and h=Φ+y∈Γ(Hom(T,E)), we have
{d+µ,Φ+y}=[d,Φ] +L Φ−L µ+ hµ,Φi .
D d Φ E
d
Since
(−L µ+ hµ,Φi )=−Φ (µ)+hµ,Φi =0
Φ E E
p d p
and α[d,Φ] = 0, we have
D
{d+µ,Φ+y}∈Γ(Hom(T,E)),
which implies that Γ(Hom(T,E)) is a right ideal of Γ(E).
On the other hand, we have
b{h,X} = b([Φ,d] +L µ−L y+ hd,ηi )
D Φ d E
d
= Φ( µ)+y(αd)=h(b(X)),
p
which completes the proof.
3 Dirac Structures and Their Reductions
Let us first study some basic properties of maximal isotropic subbundles of E. For any
subbundle Q⊂T, define:
Q0 , {h∈Hom(T,E)|h(Q)=0}.
Lemma 3.1. If rank(E) = r, dim(M) = d, then for any maximal isotropic subbundle
L⊂E, we have
rank(L )=(1−r)rank(b(L ))+r(d+r), ∀ m∈M. (8)
m m
Consequently, if r ≥ 2, both b(L) and b(L)0 are regular subbundles of, respectively, T and
E. If r =1, that is, E is a line bundle, then rank(L)=d+1.
Proof.Since L is maximalisotropic,orequivalently,L=L⊥,it is nothardtoestablishthe
following exact sequence:
0 //(b(Lm))0 a //Lm b // b(Lm) // 0. (9)
Therefore, we have
rank(L ) = rank(b(L ))+rank(b(L ))0
m m m
= rank(b(L ))+(r+d−rank(b(L )))×r
m m
= (1−r)rank(b(L ))+r(d+r).
m
6
Definition 3.2. For a vector subbundle A ⊂ T, a section s : A −→ E (i.e. b◦s = 1 )
A
is called isotropic if its image s(A) ⊂ E is isotropic. Two isotropic sections s and s are
1 2
said to be equivalent if (s −s )(A)⊂A0. The equivalence class of an isotropic section s is
1 2
denoted by s.
e
Proposition 3.3. If rankE ≥ 2, there is a one-to-one correspondence between maximal
isotropic subbundles L ⊂ E and pairs (A,s), where A is a subbundle of T and s : A → E
is an isotropic section.
e
For this reason, we call (A,s) the characteristic pair of L, and write L=L .
s,A
Proof. Let L ⊂ E be a maximal isotropic subbundle and A = b(L). By Lemma 3.1, A is
e
a regular subbundle. Any split s : A → L of the corresponding exact sequence (9) yields
anisotropic sectionand(A,s) is defined to be the characteristicpair of L. It is welldefined
since for any two isotopic sections s , s , we have Im(s −s ) ⊂ b(L)0 = A0, which is
e 1 2 1 2
equivalent to s =s .
1 2
Conversely, given a subbundle A ⊂ T and any characteristic pair (A,s), set L =
e e s,A
s(A)⊕A0. Evidently, L is a maximal isotropic subbundle of E whose characteristic pair
s,A e
is (A,s). It is also clear that if s =s , L =L .
1 2 s1,A s2,A
One may check that these two constructions are inverse to each other.
e e e
Definition 3.4. A projective Lie algebroid is a subbundle A ⊂ TM ⊕E which is a Lie
algebroid (A,[·,·] ,ρ ) and the anchor ρ =pr | .
A A A TM A
Example 3.5. LetA−→N beaLiealgebroidoverasmoothmanifoldN andαitsanchor.
Letf :M −→N be asmoothmapandf∗A→M the pullbackbundle alongf. We denote
the pull back Lie algebroid of A over M by f!A=TM ⊕ A, which is given by
TN
TM ⊕ A= (x,X)∈T M ⊕A |m∈M, and f (x)=α(X) .
TN (cid:8) m f(m) ∗ (cid:9)
Sections of TM ⊕ A are of the form:
TN
x⊕( u ⊗X ), x∈X(M), u ∈C∞(M), X ∈Γ(A),
X i i i i
such that f (x(m))= u (m)α(X (f(m))). The anchor α! of the Lie algebroid f!A is the
∗ i i
P
projection to the first summand. The Lie bracket can be locally expressed by
[x⊕( u ⊗X ),y⊕( v ⊗Y )]
X i i X j j
= [x,y]⊕( u v ⊗[X ,Y ]+ x(v )⊗Y − y(u )⊗X ).
X i j i j X j j X i i
Thus the pull back Lie algebroidf!A of the Lie algebroid A is a projective Lie algebroid in
TM ⊕f∗A.
Example 3.6. We suppose that the base manifold M is compact and let H ⊂ TM be
an integrable distribution. It is well known that there is some vector bundle E such that
the vector bundle F = H ⊕ E is trivial. Suppose that rankF = n and ε ,··· ,ε are
1 n
everywherelinearindependent sectionsofF, i.e. aframe ofΓ(F). Write ε =x +e , where
i i i
x ande aresectionsofH andE respectively. ItisclearthatΓ(H)=span{x ,··· ,x }and
i i 1 n
Γ(E) =span{e ,··· ,e } (over C∞(M)). Since H is an integrable distribution, there exist
1 n
functions ck ∈ C∞(M) such that [x ,x ] = ck x . Now set [ε ,ε ] = ck ε . It is easy to
i,j i j i,j k i j i,j k
see that F is a projective Lie algebroid in TM ⊕E.
7
A Dirac structure L ⊂ E is called reducible if b(L) is a regular subbundle of T. By
Lemma 3.1, any Dirac structure is reducible if rank(E)≥2. As a main result of this paper,
the following theorem describes the nature of reducible Dirac structures in the omni-Lie
algebroid E.
Theorem 3.7. For any vector bundle E, there is a one-to-one correspondence between
reducible Dirac structures L⊂E and projective Lie algebroids A=b(L)⊂T such that A is
the quotient Lie algebroid of L.
Proof. Assume that L is a reducible Dirac structure and let A=b(L)⊂T. Then we have
the following exact sequence:
// a // b // //
0 A0 L A 0. (10)
By L being reducible, A is a regular subbundle, A0 as well. The anchor α◦ρ vanishes if
restricted on A0. Furthermore, by Lemma 2.6 and the fact that L is a Dirac structure, A0
is an ideal of L. So we have a quotient Lie algebroid structure (A,[·,·] ,ρ ), where ρ is
A A A
clearly the projection to TM. This proves that A is indeed a projective Lie algebroid.
Conversely,fortheprojectiveLiealgebroid(A,[·,·] ,ρ ),defineasubsetLA ⊂b−1(A)⊂
A A
E by:
LA , {X ∈b−1(A) | for some X ∈Γ(b−1(A)) with X =X, there holds
m m m
e e
b X,Y =([bX,bY] ) , ∀ Y ∈Γ(b−1(A))}. (11)
n o A m
m
e e
Note that by Equation (4), we have
b fX,Y −([fbX,bY] ) =f(b X,Y −([bX,bY] ) ).
n o A m n o A m
m m
e e e e
Hence the above definition does not depend on the choice of X.
ToprovethatLA istheuniquereducibleDiracstructuresucehthattheinducedprojective
Lie algebroid is (A,[·,·] ,ρ ), we need three steps as follows. Step 1 proves that LA is a
A A
maximal isotropic subbundle such that b(LA) = A. Step 2 proves that LA is closed under
the bracket{·,·} anditfollowsthat LA is a reducible Diracstructure suchthatthe induced
projective Lie algebroid is (A,[·,·] ,ρ ). The last step proves the uniqueness of such Dirac
A A
structures.
Step 1. We prove that LA is a maximal isotropic subbundle. We will construct a
maximal isotropic subbundle L using a connection γ in the vector bundle E and prove
sγ,A
that L =LA.
sγ,A
Recall that a connection in E is a bundle map γ : TM → DE such that α◦γ =1 .
TM
Associatedwithγ thereisabackconnectionω : DE → gl(E),suchthat ◦ω+γ◦α=1 .
DE
i
So we can define a bundle map γ : E → JE by
e
hγ(e),di ,ω(d)(e)=(d−γ◦α(d))(e), ∀ d∈DE (12)
E
e
such that ◦γ =1 . In turn, we get a map:
E
p
e
γ+γ : T → E such that b◦(γ+γ)=1 . (13)
T
e e
8
We still denote this map by γ. This does not make any confusion since it depends on what
is put right after it.
Choose an arbitrary subbundle C ⊂ T, such that T = A⊕C. Define a bundle map
Ω : T ∧T → E by
γ
Ω (a,b) = [a,b] −b{γ(a),γ(b)}, ∀ a,b∈Γ(A),
γ A
Ω (c,t) = 0, ∀ c∈C, t∈T.
γ
ToseethatΩ ∈Hom(∧2T,E),firstforanya=x+u, b=y+v ∈Γ(A),wherex,y ∈X(M),
γ
u,v ∈Γ(E), we have
b{γ(x+u),γ(y+v)} = b([γ(x),γ(y)] +L γ(v)−L γ(u)+ hγ(y),γ(u)i )
D γ(x) γ(y) E
d
= [αγ(x),αγ(x)] +γ(x)( γ(v))−γ(y)( γ(u))
D
p p
= [x,y]+γ(x)v−γ(y)u,
which implies that
Ω (x+u,y+v)=([x+u,y+v] −[x,y])−γ(x)(v)+γ(y)(u). (14)
γ A
Thus we have Ω (x+u,y+v)∈Γ(E). On the other hand, for any f ∈C∞(M), we have
γ
Ω (x+u,f(y+v)) = ([x+u,f(y+v)] −[x,fy])−γ(x)(fv)+γ(fy)(u)
γ A
= fΩ (x+u,y+v)+x(f)(y+v)−x(f)y−α(γ(x))(f)v
γ
= fΩ (x+u,y+v). (15)
γ
By (14) and (15), we obtain that Ω ∈Hom(∧2T,E). We also denote the associated skew-
γ
symmetric map from T to Hom(T,E) by Ω .
γ
Define an isotropic section s : A−→E by
γ
s (a)=γ(a)+Ω (a), ∀ a∈A.
γ γ
In fact, for a=x+u, b=y+v ∈Γ(A), we have
(s (x+u),s (y+v))
γ γ E
= (γ(x)+γ(u)+Ω (a),γ(y)+γ(v)+Ω (b))
γ γ E
1
= (Ω (y+v,x+u)+Ω (x+u,y+v)+hγ(x),γ(v)i +hγ(y),γ(u)i )=0.
2 γ γ E E
By Proposition 3.3, we get a maximal isotropic subbundle L :
sγ,A
L =γ(A)+Ω (A)+A0. (16)
sγ,A γ
We can directly check that L does not depend on the choice of the connection γ and
sγ,A
the subbundle C. An alternate approach is to prove that L = LA, since LA does not
sγ,A
depend on s and A.
γ
Now we prove L = LA. Any X ∈ Γ(L ) has the form X = γ(a)+Ω (a)+h,
sγ,A sγ,A γ
where a = x+ u ∈ Γ(A) and h ∈ Γ(A0). For any Y = d+ µ ∈ Γ(b−1(A)) satisfying
9
b(Y)=y+v ∈Γ(A), we have
b{X,Y} = b({γ(x)+γ(u),d+µ}+{Ω (a)+h,Y})
γ
= b([γ(x),d] +L µ−L γ(u)+ hγ(u),di )+(Ω (a)+h)(b(Y))
D γ(x) d E γ
d
= [x,αd]+γ(x)(v)−d(u)+hγ(u),di +Ω (x+u,y+v)
E γ
= [x,y]+γ(x)v−γ(y)u+Ω (x+u,y+v)
γ
= [x+u,y+v] , (using (14) )
A
= [b(X),b(Y)] .
A
Thus,X ∈Γ(LA). SowehaveL ⊂LA. Sinceb(LA)⊂A,anyX ∈LA canbewrittenas
sγ,A
X =X +h, where X ∈L and h∈Hom(T,E). Thus h=X−X ∈LA∩Hom(T,E).
0 0 sγ,A 0
For any k ∈ Hom(T ,E ) = Kerb and k ∈ Γ(Hom(T,E)) satisfying k(m) = k, ∀
m m
Y ∈Γ(b−1(A)), we have, by Equation (7) e e
b k,Y −([b(k),b(Y)] ) = k(b(Y)).
n o A m
m
e e
Thus k ∈LA ∩Hom(T ,E ) if and only if k ∈A0 , that is,
m m m m
LA∩Hom(T,E)=A0. (17)
SowehaveprovedthatLA ⊂L . Bymaximality,LA =L andhenceLA isamaximal
sγ,A sγ,A
isotropic subbundle of E.
Step 2. We prove that Γ(LA) is closed under the bracketoperation {·,·} and it follows
that LA =L is a reducible Dirac structure.
sγ,A
For any X , X ∈ Γ(LA) and Y ∈ Γ(b−1(A)), we have {X ,X } ∈ Γ(b−1(A)) and
1 2 1 2
{X ,Y}∈Γ(b−1(A)). Moreover,we have
i
b{{X ,X },Y} = b{X ,{X ,Y}}−b{X ,{X ,Y}}
1 2 1 2 2 1
= [bX ,b{X ,Y}] −[bX ,b{X ,Y}]
1 2 A 2 1 A
= [bX ,[bX ,Y] ] −[bX ,[bX ,Y] ]
1 2 A A 2 1 A A
= [[bX ,bX ] ,bY]
1 2 A A
= [b{X ,X },bY] ,
1 2 A
whichimpliesthat{X ,X }∈Γ(LA). SoLA isaDiracstructure. InStep1,wehaveproved
1 2
that b(LA)=A, and in turn, LA is a reducible Dirac structure. By definition, the induced
projective Lie algebroid is exactly (A,[· , ·] ,ρ ).
A A
Step 3. We prove the uniqueness of such Dirac structures.
AssumethatL′ isanotherreducibleDiracstructuresatisfyingthesamerequirements. It
sufficestoprovethatL′ ⊂LA,sinceLA isamaximalisotropicsubbundle. ForanyX ∈L′
m
and X ∈ Γ(L′) such that X = X, we prove that X ∈ LA. In fact, ∀ Y ∈ Γ(b−1(A)), we
m m
are abele to find some Y′ ∈eΓ(L′) such that bY′ =bY. So we can write Y =Y′+K, where
K ∈Γ(Hom(T,E)). By Lemma 2.6, X,K ∈Γ(Hom(T,E)). Thus,
n o
e
b X,Y =b X,Y′ +b X,K =[bX,bY′] =[bX,bY] ,
n o n o n o A A
e e e e e
which implies that X ∈ LA. So we have L′ ⊂ LA. The proof of Theorem 3.7 is thus
m
completed.
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