Dirac structures and poisson reductions on poisson groupoids

Vol. 53 (2004) REPORTS ON MATHEMATICAL PHYSICS No. 1

DIRAC STRUCTURES AND POISSON REDUCTIONS ON POISSON GROUPOIDS

DE-SHOU ZHONG

Department of Economics, China Youth University for Political Sciences 100089, Beijing, China

(e-mail: [email protected])

(Received April 4, 2002 - - Revised July 23, 2003)

The notion of characteristic pairs of Dirac structures was introduced by Liu in 2000. In this paper, the invariant Dirac structures on Poisson actions and pullback Dirac structures are characterized in terms of their characteristic pairs. Poisson homogeneous spaces and Poisson reduction are discussed.

Keywords: Poisson groupoid, Poisson action, Lie bialgebroid, Poisson homogeneous. 2000 Mathematical Subject Classification: Primary 58F05; Secondary 17B66, 22A22, 53C15.

I. Introduction

Dirac structures were introduced by T. J. Courant and A. Weinstein [2]. In [I], the notion of Dirac manifolds was thoroughly investigated by Courant. A smooth Dirac subbundle on a manifold P is a smooth subbundle L of TP @ T*P such that L is maximally isotropic with respect to a symmetric bilinear form and its sections are closed with respect to bracket operation given by Courant. The manifolds with Dirac subbundles we call Dirac manifolds. The theory of Dirac structures on manifolds includes the theory of presymplectic, Poisson and foliation structures. It is a useful tool for the Poisson reduction [II]. Z.-J. Liu, A. Weinstein and P. Xu have extended the theory of Dirac structures to Lie bialgebroids [8, 6]. There is a natural relation between Lie bialgebroids and Courant algebroids. When (A, A*) is a Lie bialgebroid, A @ A* is a Courant algebroid equipped with a nondegenerate symmetric bilinear form ( , )+, a skew-symmetric bracket [ , ] on the sections of A@A* and a bundle map p. In [7], Liu, Weinstein and Xu developed some aspects of the Dirac structures of (A, A*) and Poisson homogeneous spaces. In their paper, they had to do a number of computations by hand. In [5], Liu introduced the notion of characteristic pairs of Dirac structures. A Dirac subbundle can be determined by its characteristic pairs (D, I-I), where D is a smooth distribution of A and FI ~ F(A2A) is a bivector field. Using the characteristic pairs of Dirac structures, we can give a new characterization for a series of theorems in [7]. On the way we get theorems about reduction of Poisson action.

[39]

40 DE-SHOU ZHONG

In Section 2, we recall some basic facts concerning Lie bialgebroids, Courant algebroids and characteristic pairs of Dirac structures. In Section 3, we shall discuss invariant Dirac structures on groupoids. Using the characteristic pairs of Dirac structures, we shall give new proofs of the theorems about pullback Dirac structures on the actions of Poisson groupoids. In Section 4, we shall discuss the reduction of Poisson groupoids action and Poisson homogeneous spaces by the method of the characteristic pairs of Dirac structures.

2. Preliminaries

Lie bialgebroid is a dual pair (A, A*) of vector bundles equipped with Lie algebroid structures such that the differential d. on F(A*A) coming from the structure on A* is a derivation of the Schouten-type bracket on F(A*A) obtained by extension of the structure to A. d. is a derivation for sections of A [10], i.e.

d . [X , Y] = [d.X, Y] + [X, d .Y] , VX, Y ~ F(A).

In the paper, F(A*A) and F(A) always denote the spaces of all smooth sections of the bundles A*A and A, respectively.

Let A be a Lie algebroid over P with anchor a and let A e F(A2A) be a bisection. Denote by A # the bundle map A* ~ A defined by A#(~)(r/) = A(~ e, r/), ¥~e, r/ F(A*). Define a bracket on F(A*) by

[~, rl]a = La#¢O - LA#,~ - d[A(~ , 0)].

The composition a o A # : A* --+ T P we denote by a.. Then A* with the bracket and anchor a. above becomes a Lie algebroid iff

Lx[A, A] = [X, [A, A]] = O, ¥X e F(A).

In this case, (A, A*) becomes a Lie bialgebroid automatically. When [A, A] = 0, then (A, A*) is called triangular Lie bialgebroid.

The notion of Courant algebroids was introduced by Liu, Weinstein and Xu in [6]. It was simplified by Kyousuke Uchino. Here we take the definition of Courant algebroids from [4].

A Courant algebroid is a vector bundle E ~ P equipped with a nondegenerate symmetric bilinear form ( , )+, a nonskew-symmetric bilinear operation o on F(E) and a bundle map /9 : E --~ TP satisfying the following relations:

I. x o(yoz)=(x o y)oz + yo(x oz), 2. p ( x o y) = [p(x), p(y)], 3. x o f y = f ( x o y) -t- ( p ( x ) f ) y , V f ~ C°° (P) , 4. x ox = D(x,x)+, 5. p ( x ) ( y , z)+ = (x o y, z)+ q- (y, x o z)+,

where x, y, z e F(E), D : C°° (P) ~ F(E) is the map defined by D = ½fl- lp*d, and fl is the isomorphism between E and E* given by the bilinear form.

DIRAC STRUCTURES AND POISSON REDUCTIONS 41

In a Lie bialgebroid (A, A*), there exist two natural nondegenerate bilinear forms, one symmetric and another antisymmetric, which are defined as

1 (X + ~, Y + 17)* = ~((~, Y) 4- (X, ~7)),

where ( , ) denotes operation between sections of dual bundles, X, Y e F(A), ~, ~7 e F(A*). In F(A (3 A*), the bracket operation is defined as

[X + ~, Y + 0] = (IX, Y] + L~Y - L n X - d . (X + ~, Y + 0)-)

+([~, ~71 + Lxo - Lv~ + d(X + ~, Y + 17)-).

Then E = A (3 A*, with the nondegenerate symmetric bilinear form (. , .)+, the skew-symmetric bracket [., .] on F(E) and bundle map p = a + a, : E --+ P from anchors of Lie bialgebroid (A, A*), becomes a Courant algebroid [6].

In a Courant algebroid (E, p, [., .], (., .)+), Dirac subbundle is a subbundle of E. It is maximally isotropic in the symmetric bilinear form (., .)+, and closed with respect to the bracket operation [., .]. When L is a Dirac subbundle of E, then (L, PlL, [', "]) is a Lie algebroid in general sense.

In [5], Zhang-Ju Liu used characteristic pairs of Dirac structures to characterize Dirac structures. In this paper we shall use characteristic pairs of Dirac structures to characterize Dirac structures in Lie bialgebroids.

Let (A, A*) be a Lie bialgebroid. A pair (D, H) which is composed of a smooth distribution D c_ A and a bivector field FI E r ' (AEA) corresponds to a maximal isotropic subbundle of A (3 A*,

L = {X + 1-I#~ + ~ [ ¥X e D, ~ ~ D ±} = D (3 graph(rI#1o±),

where D ± _ A* is the conormal distribution of D, D = L N A is called a characteristic distribution of L, such a pair (D, H) is called characteristic pair of L.

REMARK 2.1. Given a pair (D, H), we can get a maximal isotropic subbundle L through the method above. If L is a maximal isotropic subbundle, D = L f7 A may not have constant rank. In this paper, we assume that D = L N A has always constant rank.

THEOREM 2.1 ([5]). Let L C A (3 A* be a maximal isotropic subbundle corresponding to a characteristic pair (D, H) in a Lie bialgebroid (A, A*). Then L is a Dirac subbundle if and only if the following conditions are satisfied:

1. D is integrable distribution. 2. H satisfies the Maurer-Cartan type equation (rood D), i.e.

1 d, Fl + ~[H, rI] = 0 (modD). (1)

42 DE-SHOU ZHONG

3. D ± is closed with respect to the sum bracket [., .] + [., "]rl, i.e.

[s e, r/] + [$, r/ln e F(D±), Yse, r /e I"(D±), (2)

where [., "]n is defined as

[se, r/]n = L r i ~ / - Ln~se - d(Flse, ~/}, Vse, r / e I '(A*).

EXAMPLE 2.1. Let M be a Poisson manifold with a Poisson tensor ~r. ( T M , T ' M , Jr) becomes a triangular Lie bialgebroid. The Dirac structure L is reducible in T M ~ T * M if its characteristic distribution D = L A T M induces a foliation f , and the quotient space P = M / f is a smooth manifold such that the projection is a submersion. Let (D, rI) be the characteristic pair corresponding to L. Then, zr + FI is D-invariant (mod D) and it can be reduced to the bivector field zrt, on P.

If M is a smooth manifold and T * M has a trivial Lie bracket, then ( T M , T ' M ) becomes a Lie bialgebroid. (D, I-I) is the characteristic pair corresponding to the Dirac structure L in T M ~ T*M. The condition 1 of Theorem 2.1 is equivalent to [FI, FI] = 0 (modD) since d.Fl = 0. The third condition is equivalent to I ' (D ±) being closed w.r.t. [., "]n. For any X • I '(D), se, O • F(D±),

Lxn(se, O) = L x ( n ( s e , O)) - n(Lxse, ~) - n(se, Lxtl)

= (X, di1($, r/)) + (Lxse, fir/} - {Us e, Lxrl)

= - ( X , dFl(~, ~/)) - (X, Lnose) + (X, LrI~0)

= (x , [se, n]n) ,

the condition 2 is equivalent to LxI-I -- 0 (modD), i.e. FI is D-invariant (mod D). If D is reducible and its induced foliation 2- is, then there is a Poisson structure Pr. FI on the quotient manifold M/J: . We obtain the following conclusion.

COROLLARY 2.1. A reducible Dirac structure on a smooth manifold is one-to-one correspondence with Poisson structure on its quotient manifold.

3. Invariant Dirac structures on groupoids

Let (A, A*) and (B, B*) be two Lie bialgebroids, and let there exist a surjective morphism ¢ : (A, A*) ~ (B, B*) of Lie bialgebroids, i.e. ¢ : A --+ B is not only a surjective morphism of Lie algebroids but also a Poisson mapping, where A and B have induced Lie-Poisson structures from their dual Lie algebroids A* and B*, respectively. If ¢ is surjective, then ¢* is injective. We note that ¢* : B* ~ (ker¢) ± is a bijection, where (ker¢) ± denotes annihilator of a kernel of the mapping ¢. Therefore, there exists an inverse mapping (¢.)-1 : (ker¢)± __+ B*. We can construct a mapping

(I) = ~b I~ (q~*)-I : A ~ (ker¢) ± -+ B ~ B*.


LEMMA 3.1. Let L C B ~ B* be a maximally isotropic subbundle and L = qb-l(L) its inverse image bundle. Then L is also a maximally isotropic subbundle of A ~ A * .

Proof: Let L - - {_X+~ I X • B,~ • B*}, L - - { X + ~ I X • A,~ • (ker4))±}, where 4)*(~) = ~, 4)(X) = X, 4)*(0) = ~, 4)(Y) = Y. Then,

1 - (X + ~, Y + ~)+ = ~((X, ~) + (}', Y))

and

_

= ~ ( ( x , 4)*0) + (4)*~, f'))

1 = ~((4)(21, o) + (~, 4)(f')))

1 = ~((o, x ) + (~, r ) ) = o

dim L = dim ker 4) + dim L = dim ker 4) + dim B

= dim ker 4) + dim(ker 4))1 = dim A.

Therefore, L is maximally isotropic. []

Let a Lie algebroid A be a bundle with the base ~5. Let f be an induced map from P to the base P of the Lie algebroid B by the morphism 4) of the Lie algebroids, and f o Pr = Pro 4), T f o 5 = a o 4), where Pr, T f , gt, a denote the projection mappings of the bundles A, B the tangent mapping of f , the anchors mapping of A and B, respectively.

LEMMA 3.2. Let L C B ~) B* be a maximally isotro_pic subbundle, L = ¢b -1 (L) the inverse image bundle of L, and let (D, rI) and (D, FI)_ be characteristic pairs corresponding to the maximally isotropic subbundles L and L, respectively. Then for any ~, r /6 F(D ±) and 4)*~ = ~, 4)*0 = 0 6 F(D ±) we have the following equalities:

1. d_,(4)l=l) = 4)(d, fI); 2. [~, ~]h = 4)*[~, r/In,

where the differentials d. on F(A*A) and F(A*B) come from the Lie algebroid structures on A* and B*, respectively.

Proof: (1) For any ~, r/, y • F(B*),

d,(4) fI)(~, r/, y) = a,(~ )(4) fI(r/, y ) ) - a,(r/)(4) fI(~, y)) + a,O/)(4)I=l(~, 0))

+ 4)h(~, [0, ×]) - 4)h(r/, [×, ~]) + 4)h(z, [~, r/I).

We need to prove the following formulae for a,(~)(4)l=I(o, y)) and 4)1=I(~, [17, y]),

a,(g)(4)fI(r/, y)) o f = a,(4)*(~))(4)h(r/, ),) o f ) = 5,(4)*(~))(1=I(4)*r/, 4)*y)).

Because 4)* is injective, and ~, r/, y are all 4)*-relation sections, and 4) is a mor-

44 DE-SHOU ZHONG

p_hism of Lie bialgebroids, then ¢*[r/, y] = [¢'1/, ¢*y] and ¢1:I(~, [0, Y])o f = I-I(¢*~, [¢*0, ¢*Y]). We have

d.(¢l=I)(~, T/, y) o f = d.l:l(¢*~, ¢*r/, ¢*y) = ¢(d.l:l)(~, r/, y) o f,

i .e .d . (¢ l : l ) = ¢(d.l:l). (2) Let ¢ * ~ = ~ , ¢ * 0 = ~ and ~ be ~b related to v. Then

( ¢ ( n o , o) = (n~, ¢* , ) = n(~ , ¢*~)

= h(¢*~, ¢ * 0 = Ch(~, ~)

= ( (¢h)~ , n).

Therefore, ¢(1=I~)= (¢1=I)~ = I-I~, l=I~ is ¢ related to 1-I~,

(¢*(Ln¢rl), v} = (LnCr/, ¢(~)) = dr/(FI~, ¢(~)) + a(¢~)(rI~, ~)

= a(1-I~)(r/, ¢(~)} - (r/, [FI~, ¢~])

= a(¢(l=I~'))(rl, ¢(#)) - (r/, [¢(1=I~'), ¢~1)

= TZ o 5(1-I~)(r/, ¢(~)) - (~, [¢(FI~), ¢91)

=a(h~) (¢*~, ~) - (¢*~, [fi~, 0 )

i.e. ¢*(Ln~O)= Lf i~ . Similarly, we can prove that ¢* (Lno~)= Lr i~ ,

(¢*dn(~, ~), 0 = (an@, ~), ¢~) = a (¢9) (n(~ , ~)) = Tf(5(9))(¢I=I(~, r/)) = 5(~)(lrI(¢*~, ¢*r/))

=a(e)(fi(~, 0) = (d(h(~, O), ~), i.e. ¢*d17(~, r~)= dlq@, ~). Then we obtain

¢*[~, ~]n = [#, On. []

THEOREM 3.1. L is a Dirac structure i f and only i f L is a Dirac structure.

Proof: Let (D, I-I) be a ch_aracteristic pair corresponding to the maximally isotropic subbundle L and let (D, I7) be a characteristic pair correspondin_g to L. Then b = L A A = ~ - I ( L ) A ~ - I ( B ) = ~ - I ( L A B) = ~ - I (D) , i.e. ~ (D) = D. Since ker¢ C D, then b ± C (ker¢) ±. For _any ~_ _~ b ±, there must exist a unique

~ B* such that ~ = ¢*@). Then for any X + l'I~ + ~ ~ L we obtain

¢ ( 2 ) -q- ¢(lrl~) "~- (¢*)-1(~) ~ L,

and for any ~ ~ A*, 0 ~ B* one gets

(¢(fi~), ~) = ( n L ¢*~) = n(~-, 4,*,7)

= (¢n)(¢*-x(~-), ~)

= ((¢n)(¢*-1(~)), r~).

Hence, ¢(1=I~)= (¢fI)((¢*-1(~)) (modD), i.e. e l i = 1-I (modD).


When L is a Dirac structure, then by Theorem 2.1 the characteristic pair (D, I-I) of L satisfies the conditions of the theorem. Because ¢ : A --+ B is a Lie algebroid morphism, for any sections X, t" • I '(D) = F(L fq A), there are C-decompositions as follows [3]:

¢ o X = Z u i ( X i o f ) , ¢ o ~" = ~ oj(Yj o f )

and

¢[X, ~'] = ~ uivj([Xi , Yj] o f ) + ~_.,[X., vj](Yj o f ) - ~_,[F', ui](Xi o f ) ,

where Xi, Yj • V(D) and ui, oj are functions on the base ib of the Lie algebroid A. When D is integrable, we have [Xi, Yj] • F(D) and ¢[X, Y] can be denoted as the C-decomposition of the sections in V(D). Therefore [X, ~'] • F(D), so b is integrable.

For the second condition in Theorem 2.1, we have ¢[1=I, 1=I] = [I-I, FI] (modD) since 1:I and 1-I are C-related (modD). Let ¢ be a morphism of Lie bialgebroid and (¢,)-1 be the morphism of Lie algebroid from (ker¢) ± to B*, where (ker¢) ± is regarded as a Lie subalgebroid of A*. h. and a. are the anchors of (ker¢) ± and B*, respectively, so we have T f o it. = a. o (¢.)-1 and T f o it. o ¢ * = a.. For any ~ • F(B*), g • C°°(P) ,

T f o ?t, o ¢*(~)g ----- a, (~)g ~ h, o ¢*(~)(g o f ) = a , (~)g

and from Lemma 3.2(1), d,(¢l=I)= ¢(d, l ] ) , we obtain

1 - - d,(rl) + ~[r l , l-l] -- 0 (modD).

For the third condition, because (¢,)-1 is a one-to-one mapping, F((ker¢)±)_ consists of the sections of (¢*)-I-relation. F(D ±) C F((ker¢)±), so for any ~, ~ e r ( / ) i ) , [~, r/] = [(¢,)-1~, (~,)-1~] = (¢,)-1[@, ~], and [@, ~]fI = Lf l~ -- Lflfi@ -

d ( ~ I~,] ~(i~)e B~ Le),n~.ae. 3[~12) ~ ~ e ? ~ ~ , ~ ]~(~_~! nS .~I (D; ) wWec hanapro(v~;)t~le( [o~t~e]r

part of the theorem. []

The Dirac structure L introduced above is called the pullback Dirac structure. Since L and L are both Lie algebroids, ~ : L --+ L is also a morphism of Lie algebroids.

Now we use the concept of the pullback Dirac structure to study Dirac structures on groupoids. Let ( G ~ P , ot,/~) be a Poisson groupoid, ( T G , T*G)--a triangular Lie bialgebroid with a Lie bracket induced by the Poisson tensor zr of G on I ' (T*G) , and ( A , A * ) - - t h e tangent Lie bialgebroid of G in the sense of [13]. We construct a morphism ¢ : T*G ~ A* from the Lie algebroid T*G to the Lie algebroid A*. Let x • G, f l (x) = p, ~ • T,*G and v • Ap, then (¢(~), v) = (~, lxv).

46 DE-SHOU ZHONG

The mapping ~b is a surjection and its kernel is (TaG) x, where TaG denotes kerot.. 4~ is a morphism of Lie bialgebroid from the Lie bialgebroid (T 'G, TG) to the Lie bialgebroid (A*, A). The dual 4" of the morphism 4~ is the restricted left translation mapping which is induced by the multiplication of the groupoid by the Lie algebroid TG. We can construct the mapping

(I) = (4*) -1 (9 q~ : TaG (9 T * G -~ A (9 A * .

Let B(G) denote a Lie group which consists of bisections of the Poisson groupoid. Let E • B(G), and let the induced by E left translation mapping be denoted by Ix:. We have the following lemma.

LEMMA 3.3. The mappings Cb and ( 4 * ) - 1 introduced above are left-invariant by the action of B(G), i.e.

~b(17c(~)) = ~b(~), (~b*)-l(lxz(v)) = (~b*)-l(v), Yv • TaG, ~ • T*G.

Proof: Let v • TaG, f l ( x ) = p , y • G, f l ( y )=ot (x ) , lpcx = yx, Vvt • Ap,~ • Tv*~ G, then

(~(I~c(~)), V') = (l~c(~), IxV t) = (~, lpClxV') = (~, lyxV') = (qb(~), V').

Similarly, we can prove that (4*) -1 is left-invariant with respect to the action of B(G). Therefore, the mapping ~ is invariant w.r.t. B(G). []

PROPOSITION 3.1. Let L c TG (9 T*G be a Dirac structure. It is a pullback Dirac structure corresponding to a Dirac structure in A (9 A* if and only if" (1) L is left-invariant under B(G), (2) (TAG) ± C L.

Proof: Let L be a pullback Dirac structure of L, i.e. Z, = ~ - I (L) . Because ~ is left-invariant w.r.t. B(G) and ~(lpz(L)) = ~ (L) = L, L is left-invariant w.r.t. B(G). In addition, kerq~ = (TAG) ± and L = ~ - I (L) , then ker~ C L, i.e. (TAG) ± C L.

On the other hand, when (1) and (2) in the proposition hold, and L is isotropic, we can prove that L c TaG (9 T*G. Because L is left-invariant under B(G), then for any E • B(G), llcL = L, ~(Ll~x) = ~(llcLIx) = ~(LI~). Then ~(Llx) depends only on the base point f l ( x ) = p. We define L = ~(LIp), then L is a maximally isotropic subbundle of A (9 A*, and L = ~ - I (L) . By Theorem 3.1, L is a Dirac structure of A (9 A*. []

THEOREM 3.2. Let L c TG (9 T*G be a Dirac structure and let (D, FI) be the corresponding characteristic pair. Then L is the pullback Dirac structure corresponding to a Dirac structure in A (9 A* iff."

1. H is a lefi-invariant under B(G); 2. D C TaG is Ieft-invariant under B(G), and for all ~ ~ F(D±), I-I(~) •

r(ra ).


Proof: When L is a pullback Dirac structure, the two conditions of Proposition 3.1 hold. L is left-invariant under B(G), D = L fq TG is also left-invariant under B(G). Because

L={x+r~(#)+#I X 6 D , ~ e / ) ±}

and the equality Ix:(l=l~)= (l~zH)(l[_~) holds for any /C e B(G), and TG/D, D ±

is also left-invariant under B(G), then

l~zL = {l~zX + I~(FI~) + l~z_,~ }.

l~zX E D, I~_1~* - E 1)± and l~(l=l~) = 1=I(I~_~), i.e. l~zl=l = l=l(modD). Therefore, 1=I

is left-invariant under B(G). Since (TUG) ± C L, ¥~ e F((T~G)±), Y + 1=I~ + ~ F(L), ~ e F(/9±), we have

I=o,

i.e. Y+H~ ~ F(T~G). When ~----0, then Y ~ F(T~G), and b C TUG. Taking Y = 0, we have FI(~) ~ F(T~G).

On the other hand, when _(I), (2) of the theorem hold then D and l=I are left-invariant under B(G), i.e. L is left-invariant under B(G).

For any ~ E F(/)±), I'I(~) ~ F(T~G), we know that D C TUG. Then we have F(/)) + I=I(F(/)±)) C F(T~G), i.e. p(Z) c T~G. Hence (TUG) ± C p(L) ± = LMT*G C L, where p denotes the projective mapping of L ~ TG. By Proposition 3.1, L is a pullback Dirac structure. []

When L is a reducible Dirac structure and ~" is the foliation corresponding to /), then G/J r is a Poisson manifold. Let L be the pullback Dirac structure of L, and let the characteristic subspace D of L be a Lie subalgebroid of A. When the integral manifold of D is a-simple connected, connected closed subgroupoid, then L is called normal Dirac structure. We obtain the following corollary.

COROLLARY 3.1. Let L be the pullback Dirac structure of L, then L is a reducible Dirac structure iff L is a normal Dirac structure.

4. Poisson homogeneous spaces

If the action of a Poisson Lie group on a Poisson manifold is transitive, then the Poisson manifold is called Poisson homogeneous manifold. We know that the Poisson structure of the manifold is not preserved by the Poisson action. Hence, the homogeneous Poisson manifold represents a "hidden symmetry". Because the action of Poisson groupoids on Poisson manifolds is not complete, it is not transitive in the usual sense. Any homogeneous space is isomorphic to its action group modulo an isotropic subgroup. The homogeneous spaces under the action of groupoids are also isomorphic to this groupoid modulo a wide subgroupoid [9].

DEFINITION 4.1. Let (G:::~P,a, ~) be a Lie groupoid, M - - a manifold and a G-space with a moment mapping J : M ~ P. If there exist sections tr of J such

48 DE-SHOU ZHONG

that G . t r ( P ) = M, then M is regarded as a homogeneous space, with isotropic subgroupoid H = {h ~ G I h . t r (P) C tr(P)}.

Such sections were called saturating sections in [6]. Liu, Weinstein and Xu proved the following theorem.

THEOREM 4.1 ([6]). A G-space is a homogeneous space iff it is isomorphic to G modulo a wide subgroupoid H, i.e. it is isomorphic to G / H .

In this paper we use the concept of the characteristic pair of Dirac structures to give a theorem about Poisson homogeneous spaces. Let G be a Poisson groupoid with a Poisson tensor Jr~, then (TG, T ' G , Jrq) constitutes a triangular Lie bialgebroid. Now we assume that Dirac structure L C T G ~9 T*G is reducible, and its corresponding characteristic pair is (h, I-I). 7-/ denotes the foliation corresponding to h. If a Poisson action of a Poisson groupoid on a Poisson manifold can induce a Poisson action of the groupoid on the quotient Poisson manifold, we call this action a reduced Poisson action.

THEOREM 4.2. The action o f G on G/7-[ is a reduced Poisson action iff L is a pullback Dirac structure.

Proof: When the action of G on G/7-[ is a reduced Poisson action, 7-[ is invariant under the action of G, i.e. h is left B(G)-invariant; h C TUG and 7-[ C u-fiber. Let the projective mapping of G--~ G/7-( be Pr, and the Poisson structure on G/7-[ be zr. z~ is the lift of a" to G. The action of G on G/7~ is Poisson, so

zt(g[x]) = lgTr([x]) + r[xlTrG(g) -- rix]IgzrG(u),

where f l ( g ) = J ( [ x ] ) = u. Hence, for the lifted Poisson structure ~ we have

and

f r (gx) = lg~(X) + rxzr6(g) - rxlsrcc(u) (mod h)

(fr - rcc)(gx) = fr (gx) - ZrG(gX) (mod h)

= (lg~(X) + rxzrc(g) - rxlgTrG(U))

--(IgTrG(X) + rxrC6(g) -- rxlgTrG(U)) (mod h)

= lg~(X) - lgTrG(X ) (mod h)

= Ig(~ - 7rG)(X) (mod h),

i.e. H = ~ - zr~(mod h) is left B(G)-invariant. Because J o Pr = a and r[xlg = g[x] = [gx] = Pr(gx) = Prrx(g) , for any ~o e Coo(G/7-O, f ~ C°°(P) , we have

and

X j . f [x ]d9 = (r[x]). Xc~. f (u )d9

~(Pr* d(J* f ) , Pr* d~o)(x) = zr~(ot*df, r~x]d~o)(u ) (mod h),


fr(ot*df, Pr* dg) (x ) = Jr~(ot*df, r* Pr* dg)(u) (mod h).

Since G is a Poisson groupoid and the multiplication G x G --+ G of G is a Poisson action of G on itself, then

zrc(ot*df, Pr* dg)(x ) = zr~(a*df, r* Pr* dg)(u) (mod h),

i.e.

and

and

fr(ol*df, Pr* dg)(x ) - zr6(ot*df, Pr* dg)(x) = 0 (mod h)

FI(u*df, Pr* dg)(x ) = (~ - zrc)(a*df, Pr* dg)(x ) = 0 (mod h)

(H(Pr*dg),ot*df} = 0 (mod h),

then H(Pr*d9) ~ T'~G. By Theorem 3.2, L is a pullback Dirac structure. We shall prove the other part of the theorem by inverting the above process. []

COROLLARY 4.1. I f The conditions are the same as in Theorem 4.2, and when is a wide closed subgroupoid of G, then G/7-[ is a Poisson homogenous space

iff L is a pullback Dirac structure.

Let (G=~P, ot,/~) be a Poisson groupoid, M a Poisson manifold, and also a Poisson G-space with the moment map J : M --> P. We have proven that there is a morphism (TM, T ' M ) ~ (A, A*) of Lie bialgebroids, where (A, A*) is the tangent Lie bialgebroid of the Poisson groupoid G, A is regarded as N(P , F) = r e r - r_Lr_ - - TpP = U Tu~-l(u) , and A* is regarded as N*(P, F) = T~P = U u ,- (u)).

uEP uEP The anchor a of A is c~,, and the anchor a, of A* is ~rG [13]. We construct the morphism 9 : T*M ---> A* such that for any ~ e T ' M , and XJ(m) ~ A J(,, O,

(9(~), Xj~m)) = (~, XM(m)},

where Xm(m) ~ TmM denotes the infinitesimal generator corresponding to Xj(m). The annihilator of ker9 of this morphism at every point on M is the subspace of the infinitesimal generator on M under the action of Lie algebroid A. There- fore, A --> (ker9) ± is a pointwise bijection. There is a pointwise inverse mapping (kerg) ± --+ A, denoted by (9*) -1. We get the map

(I) ~--- 9 "~- ( 9 * ) - 1 " T*M 6) (ker 9) ± ~ A* 6) A.

LEMMA 4.1. Let L C A*6) A be a maximally isotropic subbundle and L = • - I (L) be an inverse image bundle of L, then L is also a maximally isotropic subbundle of T* M 6) T M.

The proof of this Lemma is similar to that of Lemma 3.1. By Theorem 3.2, when L is a Dirac subbundle, L is also a Dirac subbundle. In

this case L is called a pullback Dirac structure of L. When L is a reducible Dirac

50 DE-SHOU ZHONG

subbundle, D = L rl T M = L th (ker ~o) ±, )r is the foliation corresponding to D, then M/J r is a Poisson manifold. Using the arguments similar to Lemma 3.3, we can prove that the map qb is invariant under the action of the groupoids. To associate this with Proposition 3.1, and Theorem 3.2, we have the following theorems.

PROPOSITION 4.1. Let Z c T*M ~ T M be a Dirac structure. It is a pullback Dirac structure of a Dirac structure in A ~ A* iff." (1) L is G-invariant; (2) ker~0 C L.

THEOREM 4.3. Let Z c T*M ~) T M be a Dirac structure, and let corresponding characteristic pair be (D, FI). Then Z is a pullback Dirac structure of a Dirac structure in A ~ A* iff."

(1) H is G-invariant; (2) b C (ker~o) -L is G-invariant, and FI(~) ~ F(ker~0) ± for all ~ ~ F(D±).

By the theorem above, when Z c T*M ~ T M is the pullback Dirac structure of L C A ~ A*, then the corresponding characteristic subspace D is G-invariant, and D is left B(G)-invariant and invariant under the right translation. On the same orbit, D are isomorphic with each other at every point and /) are isomorphic with each other at J - l ( [u]) . Suppose L is reducible, then by Theorem 3.2, L is also reducible. Their corresponding foliations are )r and 7-t, and quotient manifolds are M/~: and G / ~ , respectively. The action of the Poisson groupoid is induced by the moment map J , and the moment map of G x M ---> M cannot induce any action of G / ~ on M/J r . When /3 C ker J, , then D belongs to the tangent Lie algebroid of the isotropic subgroupoid of the Poisson groupoid G. We obtain the following theorem.

THEOREM 4.4. Let Z c T*M ~ T M be a reducible Dirac structure with the corresponding characteristic pair (D, H), the foliation jr and /) C ker J, , V~ F(D-L), l=I(~) e ker J,. When L is a pullback Dirac structure, then the action of G on M/J r induced by the action of G on M is also a Poisson action.

Proof: Let the Poisson tensor of Q = M/~: be 7/'Q and let 7~Q be the lift of ZrQ to M , then l=I = 7~Q - - zr(mod/9). Let IC ~ B(G) be a bisection through the point g in G and 3; be a section of J through the point x in M. First we prove that the action of G on M/T_ is defined well. The moment map of M/J r to G is J and J[x] = J(x) , l~ (g )= J[x] = J ( x ) = u, o t (g )= v. We define the action of G on M / ~ as

G x M/J r ---> M/~ r, (g, [x]) ---> g . [x] = [gx]

Let 7-[ be the foliation of D, then 7-/[e belongs to isotropic subgroupoid of G since b C ker J,. For x, y ~ M, x -,- y or y ~ [x], there exists h a 7-tu such that y = h.x. Because D is (Adx:),-invariant, 7-t is Adx:-invariant. Therefore,

g(hx) = (gh)x = (ghg-1)(gx) ,


where ot(ghg -1) = or(g) = ~ (ghg -1) = v, ghg -1 = Adx: h e 7-/u. Hence, (Adx: h) (gx ) e [gx], i.e. gy e [gx]; conversely, if gy ,-~ gx, there is h' ~ 7-/~ such that gy = h ' . (gx) , g-1 . (gy) = g - l ( h , g x ) = ( g - l h , g ) x = Ad~z_l hlx, y = AdKz-J h'x and Adx:-~ h' e ~u. Hence y ~ x and g • [x] ---- [gx] is defined well. Also, rtx]g = g[x] = [gx] = Pr(rxg).

When L is a pullback Dirac structure, the characteristic pair of L satisfies Theorem 4.3. Therefore,

(1) l=I is left B(G)-invariant, i.e. l~:l=I = l=I (rood /)), so we have

l lc(~a(X) - z r ( x ) ) = lxz ( - l (x ) = ( - l ( g x ) = ~ a ( g x ) - - 7 r ( g x ) (mod D) = fgO(gX) -- (l~zr(x) + ryzrG(g) -- ryl~zZrG(U)) (mod D).

Hence, ~Q(gX) = l ~ r a ( X ) + rylrG(g) -- rylx:rG(u) (mod D). ( ,)

We note that 3: is the section of J through the point x, for [x] C J - l ( u ) , P r . y is also the section of J through Ix]. Pr : M --+ M / 3 r is projective mapping, and

P r ~ a = ~ra, Pr(x) = [x],

With the help of Pr, the formula ( .) becomes

P r y = [y].

Zra(g[x]) = l~:rQ([X]) + r[y:rc(g) -- r[yllpczrG(u).

(2) For any ~b e COO(M/F ), Pr*d~b e F(D ±) and f ~ C°°(P) , by the above l=l(Pr* d40 e ker J,, and Pr* J* = J*, so we obtain

(I](Pr* d4,), Pr* d O * f ) ) = O,

i.e.

((Z~Q -- yr)(Pr* dqb), Pr* d(~l*f ) ) = 0 (mod D),

(~Q(Pr* d~b), Pr* d C l * f ) ) = Qr(Pr* d$) , Pr* d(~l*f ) ) (mod /3),

(~Q(Pr* d C l * f ) ) , Pr* d~b) = (yr(Pr* d ( J * f ) ) , Pr* d~b) (mod 3 )

= Q r ( d ( J * f ) ) , Pr* dq~) (mod /))

= X j . f ( x ) ( P r * dqb) (mod 3) .

Because G × M ~ M is a Poisson action,

We obtain

(r~),X,:y(U) = Xs , : (x ) .

(Tra(Pr* d O * f ) ) , Pr* dq~)(x) = (rx) .X~. f (u) (Pr* dd~) (mod /)).

52 DE-SHOU ZHONG

Hence,

i.e.

7raCl* f)[x] = (r[x]),X~,f(u),

X j , f [x] = (rtxl),X~,f(u).

By the proofs of (1), (2) and Theorem 7.1 [7], we prove that the action is Poisson. []

COROLLARY 4.2. Let L c T*M ~9 T M be a pullback reducible Dirac structure, (i), FI) be the characteristic pair corresponding to L, and V~ • F(D±), 1=I(~) • ker J,, and D C A be the pullback of D. Suppose that D is the Lie algebroid of the isotropic subgroupoid H of G, then the Poisson action of G on M can be reduced to the action of G on M / ~ and this action is also Poisson.

COROLLARY 4.3. Under the same conditions as in Corollary 4.2, when the projective mappings Pra " G --+ G / H and PrM : M ~ M / E are Poisson, the action of the quotient groupoid G / H on the quotient manifold M / J Z is Poisson.

Proof: The action of the quotient groupoid G / H on the quotient manifold M / ~ is defined as

G / H x M/Y r --+ M/J r, ([g], [x]) ~ [g][x] = [gx], fl(g) = J(x).

The definition is reasonable, since fl[g] = fl(g) and J[x] = J(x). The source and the target maps on G / H will be denoted by & and ~, respectively, then & o P r a = ~ and f l o P r o = f t . For g ' • [g], y • [x], there exist h ,h ' in H such that g' = gh, y = h' • x, then

g' . y = (gh) . (h' . x) = g(hh' . x) e g . [x] = [gx].

Therefore, the definition is reasonable. G / H is a Poisson groupoid, as illustrated by the following commutative diagram:

Poisson G x G , G

Pra x Pra I J Pra

a l H x O I S • o I n

The diagram is defined for multiplicative elements. By Proposition 7.1 in [12], G / H x G / H -~ G / H is a Poisson mapping. Similarly, we can also use the following

DIRAC STRUCTURES AND POISSON REDUCTIONS

c o m m u t a t i v e d i a g r a m to i l lus t ra te the ac t i on o f G / H o n M / : ,

P o i s s o n G x M • M

P r c x PrM PrM

53

G/H x M / ~ • M/.~

By Proposition 7.1 in [12], if G / H x M/~---> M/.~ is a Poisson mapping, then the action of G / H on M / ~ is Poisson. We complete the proof of this coronary. []

Acknowledgments I would like to thank Zhang-Ju Liu and Long-Guang He for guidance and

encouragement. I would also like to thank Ping Xu, K. Mackenzie and Jiang-hua Lu for helpful comments.

REFERENCES

[1] T. J. Courant: Trans. A.M.S. 319 (1990), 631. [2] T. J. Courant and A. Weinstein: Seminare sud-rhodanien de gbombtrie VIII. Travaux en Cours 27,

Hermann, Paris, 1988. [3] P. J. Higgins and K. Mackenzie: J. Algebra 129 (1990), 194. [4] Kyousuke Uchino: Lett. Math. Phys. 60 (2002), 171. [5] Z.-J. Liu: Banach Center Publ. 51 (2000), 165. [6] Z.-J. Liu, A. Weinstein and P. Xu: Differential Geometry 45 (1997), 547. [7] Z.-J. Liu, A. Weinstein and P. Xu: Commun. Math. Phys. 192 (1998), 121. [8] Z.-J. Liu and P. Xu: Geom. Funct. Anal. 6 (1996), 138. [9] K. Mackenzie: Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge University Press,

Cambridge 1987. [10] K. Mackenzie and P. Xu: Duke Math. J. 18 (1994), 415. [11] J. E. Marsden and T. Ratiu: Letr Math. Phy. 11 (1986), 161. [12] I. Vaisman: Lectures on the Geometry of Poisson Manifolds, Birkh~iuser, Basel 1994. [13] A. Weinstein: J. Math. Soc. Japan 411 (1988), 705.

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Dirac structures and poisson reductions on poisson groupoids