- Email: [email protected]
Almost homogeneous Poisson manifolds
VO1. 56 (2005)
REPORTS ON MATHEMATICAL PHYSICS
No. 1
ALMOST HOMOGENEOUS POISSON MANIFOLDS QILIN
YANG
Mathematical Department, Tsinghua University, 100084, Beijing, P. R. China (e-mail: qlyang @math.tsinghua.edu.cn) (Received February 10, 2005 - Revised May 9, 2005)
We prove that any connected holomorphic Poisson manifold has an open and dense symplectic leaf which is a pseudo-Kiihler submanifold, and we define a new obstruction to study the equivariance of momentum map for tangential Poisson action. Some properties of almost homogeneous Poisson manifolds are studied and we prove that any compact symplectic Poisson homogeneous space is a toms bundle over a dressing orbit. Keywords: tangential Poisson action, almost homogeneous, momentum map. 2000 Mathematics Subject Classification: primary 53D17, 53D20; secondary 58F05, 58H05.
1.
Introduction
Motivated by the study of the Poisson nature of dressing transformation for soliton equations, Lu and Weinstein introduced the notion of Poisson action [12, 13, 17]. In the Poisson geometry a surprising fact is that Poisson action does not preserve Poisson structure in general, and it preserves the Poisson structure if and only if the Poisson Lie group has zero a Poisson structure. In this way Poisson actions are used to understand the "hidden symmetries" of certain integrable systems. Since Poisson manifold generalizes the concept of symplectic manifold, it is natural that the Poisson action generalizes the concept of symplectic action, and, in particular, the Hamiltonian action on symplectic manifold. Tangential Poisson action G × P --~ P is a special kind of Poisson action which is the most close generalization of Hamiltonian action. The orbits of tangential Poisson action locate in the symplectic leaves of P and the restricted action of G to every symplectic leaf is also a Poisson action, cf. [18]. Momentum mapping for a Hamiltonian action is the abstraction of momentum and angular momentum, which are associated with the translational and rotational invariance of the potential function of a classical mechanical system. The symmetry of the potential function leads to conservation of some quantities. This allows to simplify the description of mechanical system. In 1989, Lu generalized the concept of momentum map from Hamiltonian action to tangential Poisson action and studied the Meyer-Marsden-Weinstein momentum reduction for tangential Poisson action on symplectic manifold in regular case [12]. Later on, Ginzburg investigated systematically the existence and uniqueness of [93]
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equivariant momentum map for tangential Poisson action in [7] via calculating the Poisson cohomology. Recently we generalized the momentum 'reduction to singular value level for general tangential Poisson actions in [18]. In this paper, we discuss other properties of the momentum mapping for tangential Poisson actions. The paper is organized as follows: in Section 2, we give a detailed description of symplectic stratification of a Poisson manifold and emphasize the symplectic stratification of a holomorphic Poisson manifold. In Section 3, we survey the known dispersed results on the classifications of Poisson Lie groups and investigate Poisson Lie group structures on Abelian Lie groups which have been not considered so far. In Section 4, we give the necessary and sufficient conditions for an action to preserve the Poisson structure, and define an obstruction in describing the equivariance of momentum map for a general Poisson action which does not necessarily preserves the Poisson structure. As far as I know, all known until now obstructions were studied for some special kinds of Poisson actions when the Poisson structures were invariant under such actions. In Section 5, we prove that the compact symplectic Poisson homogeneous space is a torus bundle over some dressing orbit, generalizing similar results for Hamiltonian actions given in [8]. At the end, for the first time, we give an example of nontransitive tangential Poisson action.
2.
Holomorphic Poisson manifolds and their symplectic leaves A (real) n-dimensional differentiable manifold P is called a Poisson manifold if the set of smooth functions C ~ ( P ) is a Poisson algebra over real number field R. If
we denote the Poisson bracket of functions f and g by {f, g}p, then the equivalent definition is that {f, .}p is a derivation on the commutative algebra C ~ ( P ) and meanwhile ( C ~ ( P ) , {, }~) is a Lie algebra over JR. We can conclude (cf. [16]) that there exists a bivector Jrp c F(A2Tp), called Poisson bivector field, such that the Schouten bracket [:rp, 7rp] = 0, which just corresponds to the Jacobi identity for Poisson bracket. Let ~z~ • F(T*P) --~ F ( T P ) be the bundle map defined by (:r~(/~), a) = zrp(a, I3). Then X I := -Tr~(df) : { , f } is a vector field, called the Hamiltonian vector field associated with function f. The Poisson structure induces a Lie algebroid structure (cf. [11]) on the cotangent bundle T*P. The Lie bracket on the space F ( T * P ) of 1-forms, still denoted by {, }p, is given by (cf. [12, 16]): {oe, 15}p = d(a'p(o~,/3) - (;v~ (oe), d]5) + (n'~ (/3), doO,
(2.1)
if V E I'.(TP) is a vector field, then (V, {a, fl}p) = (Lv:rp)(a, t ) - (:rr~(a), d(i(V)fl)) + (:'r~(fl), d(i(V)a)).
(2.2)
For a Poisson manifold P which is a complex manifold, clearly the algebra of complex-valued smooth functions C ~ ( P , C), coinciding with the base extended algebra C ~ ( P ) ® ~ C, is in a natural way a complex Poisson algebra. This fact tell us that studying complex Poisson algebra of smooth functions is to some extent
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reduced to the case of real Poisson algebra of smooth functions. In other words, the classifications of smooth Poisson structures on real and complex manifolds have no essential differences. To stress the impact of the complex structure on the Poisson structure :re, we introduce the following definition. DEFINITION 2.1. A complex manifold P with Poisson bivector field :rp c F(A2TI'°P) is called a holomorphic Poisson manifold, if :rp is also a holomorphic bivector field, where TI,°P denotes the holomorphic tangential bundle of P. Similarly to the real case, it is very easy to verify that a complex manifold P is a holomorphic Poisson manifold iff the algebra O(P) of holomorphic functions is a complex Poisson algebra. We remark here that our definition of holomorphic Poisson manifold coincides with that in [4, Definition 3.1], where it is called holomorphic complex Poisson structure. However, in [4] the authors investigated mainly almost complex Poisson manifold, which means that the underlying manifold is only an almost complex manifold. Hence what they called almost complex Poisson manifolds are more general objects than holomorphic Poisson manifolds considered here. For any Poisson manifold P of real dimension n, if the Poisson bivector :re is a nondegenerate tensor, the inverse bundle map of :r~ thought of as a two-form is a symplectic form [16], therefore P itself is the unique symplectic leaf. However, in general case, :re may have varying rank. Let
D = {V • r ( T P ) I 3 f e C°°(P), X f ( p ) = V(p), Vp E P}.. D is a differential distribution on P. By Jacobi identity we have [X f, Xs] = X{I,~}. So D is an involutive distribution. In fact, it is a completely integrable distribution (cf. [16, Theorem 2.6]). We denote the integral leaf through p by .Tp. Since the Hamiltonian flows defined by Hamiltonian vector fields preserve the Poisson brackets, the restriction of :rp to .Tp has constant rank. In this way we know that ,Tp is a symplectic submanifold of P and hence a symplectic leaf of P. If P is compact, let Ham(P) C Diff(P) be the Hamiltonian diffeomorphism group with Lie algebra { X f l f • C ~ ( P ) } . Then the symplectic leaves are just the connected components of Ham(P)-orbits in P. Let r(p) := dimRffp, then r(p) is equal to the rank of :rp at p. Since Jr/, is skew symmetric, r is an even integer-valued function bounded by the dimension of P. Naturally, P is decomposed into the union of symplectic leaves with varying dimensions,
U
peP O
If moreover P is a holomorphic Poisson manifold, the above decomposition has subtle structure. Let m denote the complex dimension of P and thus the real dimension of P is n = 2m. Let (Zl, z2 . . . . , Zm) be a local holomorphic coordinate system at p • P, and write :re(P) = Ei,j :retj 8zi /X 8zj. For a positive integer k < m and two k-tuples Ik = (il, i2 . . . . . ik) and Jk = (jl, j2 . . . . . jk), let :r/k& be the
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ij " " submatrix of (rOe)toxin, whose entry at itx-th row and jv-th column is ZrpuJ~, here 1 < /z, v < k. Let det(zr~~&) denote the determinant of the k-order square matrix Ik4 Jrf , and define the functions
dk = ~ Ldet(@k4)12, Ik,J~
k = 1,2 . . . . . m,
ij where summation runs on all k x k submatrices of (Zrp)mxm. We still use r(p) to denote the real dimension of a symplectic leaf passing through p e P, and denote 'max' the maximum of r. Let PI = {P e P I r(p) = l} and QI = {p e P I r(p) < l}. Then Pmax is the union of symplectic leaves of P where the Poisson structure has maximal rank and m
P = U P21 = Qmax. k=l
PROPOSITION 2.1. I f P is a holomorphic Poisson manifold, then for any positive integer l, the set P2l-1 is empty; if Q2l is nonempty, then the real codimension of the subset Q2t-2 in Q2l is at least 2.
Proof: Clearly P2l-1 is empty since the rank of a skew symmetric matrix is always even. We have l
Q21 = U P2k = {P E P I dl+l(p) = dl+2(p) . . . . .
din(p) = 0},
k=l
where the last equality follows from the definition of the rank of a matrix. Therefore we have Q2l-2 = {p E Q21 I d l ( p ) = 0}. Since locally all determinants of submatrices det(zrfkJk) are holomorphic functions on P for any k-tuple Ik and Jk, all Q2~ are analytic subsets of P. If Q21 is nonempty, then there exists at least one nondegenerate submatrix Jr) J~ of l-th order. Note that
Q2l-2 = {P ¢ Q21 I dr(p) = O} C {p E Q21 I det(rcY l) = O} and d e t U r y z) is a nonzero holomorphic function, thus the real codimension of Q21-2 in Q21 is at least 2. [] COROLLARY 2.1. For a holomorphic Poisson manifold P, if P2z is nonempty, then P2z is a relative open subset in Q21. In particular, Pmax is an open connected density subset of P if P is a connected holomorphic Poisson manifold.
Proof: From the proof of Proposition 2.1 we know that Q2z = P2l u Q2l-2. Since Q21 and Q2z-2 are closed subsets, by Proposition 2.1, P2I is a relative open subset in Q2z. Moreover Pmax is the complement of the closed subset Qmax-2 in Qma~ = P, and Qm~x-2 is a subset in Qmax of a real codimension equal at least to two. Note
ALMOST HOMOGENEOUS POISSON MANIFOLDS
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that any subset of real codimension two does not destroy the connectedness of a manifold, so Pmax is an open connected dense subset of P. [] Any smooth Poisson manifold P of real dimension n has the rank decomposition P = U0_ 0, and f ( x , y) = x when y < 0. Note that here Pmax ---- {(x, y) E ~ 2 ] X ~ 0, y < 0} is the symplectic leaf where the rank of rre is 2, it is an open but not a dense subset of ~2). Since Hamiltonian flows preserve the rank of the Poisson bivector field, all P2z are Ham(P)-invariant subsets of P (assume the P is compact when Ham(P) is not appropriately defined). Denote the connected components of Ham(P)-orbits by ~c2, where the index k indicates that the dimension of 5c2 is 2k, and the index )~ parameterizes the connected components of symplectic leaves in Pzk. Denote by Ak the set of symplectic leaves with dimension 2k, thus Pzk = UzeAkf2. Therefore, for any real n-dimensional Poisson manifold P we have a symplectic decomposition
U
)~A k, l
f;"
The symplectic leaves in Pmax are all open. So, the Ham(P)-action has at least one open orbit. If (P, :re, J) is a holomorphic Poisson manifold with complex structure J, the Hamiltonian vector field of any holomorphic function is a holomorphic vector field. So all symplectic leaves are complex symplectic submanifolds and the restriction of rrp to z the symplectic leaf Y:~ defines a symplectic form wkz satisfying cok(J., J.) = cokz (., .). Therefore all symplectic leaves 5c2 are pseudo-K~ihler submanifolds of P. Thus we have the following result. PROPOSITION 2.2. Let P be a holomorphic Poisson manifold of complex dimension m. Then we have a symplectic decomposition
t,=
U
~.~A k, 1 < k < m
where all symplectic leaves ~ are pseudo-Kiihler submanifolds of P. Moreover, P has at least one open symplectic leaf which is dense in P if P is connected. 3.
Poisson Lie groups
A smooth map between smooth Poisson manifolds M and N is called a Poisson map if the pull-back map f * : C ~ ( N ) -~ C ~ ( M ) is a Poisson algebra homomorphism. Given two Poisson manifolds (M, r~M) and (N, rCN), there is a unique Poisson structure, denoted by JrM G row, on the product manifold M × N, such that the projections from M × N to each factor M and N are both Poisson maps. We called it the product Poisson structure. A Lie group which is at the same
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time a Poisson manifold is called a Poisson Lie group if the multiplicative map lz : G x G --+ G, (g, h) --+ gh is a Poisson map, where G × G is equipped with the product Poisson structure. Let rg and lg respectively denote the left and right translation in G by g e G. Then the Poisson bivector field zrG satisfies rcG(gh) = Igzr~(g) + rhZrG(g),
Vg, h c G,
(3.1)
and is called a multiplicative bivector field. If b o t h g, h are specialized to be the unit e of G, then zrG(e)= 0. So a Poisson Lie group can never be a symplectic manifold. Let 3r~(g) = rgrCG(g). Then rc~(gh) = 7r~(g) + Adg zr~(h),
Vg, h e G,
so the linearization of zr~ at the unit is a A2~[ -valued 1-cocycle on g = Z;ie(G) relative to the adjoint action of g on A21~, that is 6 = dezr~ e H 1(~1, A2g). If 3 is a 1-coboundary, then G is called a coboundary Poisson-Lie group. The dual map 6* defines a Lie algebra structure on the dual space g* of g. There is a Lie algebra structure on g @ g* such that both g and g* are Lie subalgebras. (g ® g*, g, ~t*) is called the Lie bialgebra of (G, zrG). There is a 1-1 correspondence between connected and simply connected Poisson Lie groups and Lie bialgebras (cf. [13]). The connected and simply connected Lie group G* with Lie algebra g* is also a Poisson Lie group, called the dual Poisson Lie group of G. A Lie group D ( G ) with Lie algebra ~I @ g* is called a double for the Poisson Lie group G if the multiplication map G x G C D ( G ) , (g, h) ~ gh is a diffeomorphism. In this case, the left-action of D ( G ) on itself induces an action on G*, restricting this action to subgroup G we obtain an action G x G* --+ G*, (g, h) ~-> D l (g), it is called the left dressing action of G on G*. If G has trivial Poisson Lie group structure, then G* = g* has Lie-Poisson structure, and the left dressing action is just the coadjoint action of G on g*. The definition of right dressing action is likewise (cf. [12]). If G is semisimple, the Whitehead theorem says that both Hi(g, V) and H2(I~, V) vanish for any ~t-module V. Thus every connected semi-simple Poisson Lie group is coboundary Poisson Lie group. The 1-cocycle is given by 3 (X) = adx A for some _A E A2t~. By integration we get a 2-cycle ZrG(g) = lgA - r g A on the group G. Clearly zr6 is a Poisson bivector if and only if [A, A] e A3g is Adc-invariant. Such a A is called an r-matrix. Hence all Poisson Lie group structures on semisimple Lie groups are defined by r-matrices. Let G = T m x R n be a direct product of m-dimensional torus group and n-dimensional vector group IRn and let zr~(01 . . . . . Om, xl . . . . . x , ) =
--
G~(eiO1 . . . . . e i°m,xl . . . . . Xn)
7"g r
(3.2)
be the lift of 7r~ on the universal covering space N m+n. By the multiplicative condition (3.1), we know that (~rc)i j is a linear function with respect to the variables 0k and xl and periodic with respect to Ok. So the Poisson bivector has
ALMOST HOMOGENEOUS POISSON MANIFOLDS
the form
99
m-}-n
: } 2 C/ u 0.i A 0.j,
(3.3)
i,j,k=l
where the C/~ are the structure constants of an (m + n)-dimensional Lie algebra and C~. = 0 for k < m; ui =Oi for 1 < i < m and Um+i = x i for 1 < i < n. In particular, if G is the torus T m, is has only zero Poisson Lie group, thus any Poisson Lie group structure on compact abelian group is trivial. If G is the vector group IRn, the nontrivial Poisson Lie group structure on it must be linear and is called a Lie-Poisson structure. For a multiplicative abelian group G = (R+) n or (C*)", the solution of (3.1) is of the form n
(zr;)u~(zl . . . . . zn) = ~
C ~ lnz~,
(3.4)
k=l
rrc is a Poisson structure iff C ~ = - C ~ . for 1 < / x , v, 8 < n and n Z
/z
v
/x
v
p,
v
[(Cp~ C8× + Cp~,C~×) In zu In z~ + Cp, C~y In zu] + c.p. (p, 8, Y) = 0,
(3.5)
/z,v=l
where 1 < p, 8, F --- n, and c.p.(p, 8, F) means cyclic permutation with respect to p, 8 and g. After some simple calculations, we know that (3.5) is equivalent to n Z
/z v /x v /z v (Cp~C~, + C~,C×p + Cy~Cp~) = 0,
1 < Ix, p, 8, g <- n.
(3.6)
v=l
So Cp~ are also the structure constants of an n-dimensional Lie algebra. Therefore the Poisson Lie group structures on noncompact abelian groups are essentially equivalent to the Lie algebra structures. Belavin and Drinfeld classified all Poisson Lie group structures on complex simple Lie groups [2], Levendorskii and Soibelman obtained the classification of Poisson-Lie' group structures on compact connected Lie groups in [10]. The classification of Poisson-Lie group structures on a non-reductive Lie group is more involved. The cases of 2-dimensional affine group, Poincar6 group and Galilei group were solved completely in [3, 14, 19].
4.
Poisson actions and momentum mappings
An action of a Poisson Lie group (G, zrG) on a Poisson manifold (P, rre) is called a Poisson action (cf. [13, 15]) if the action map ~r : G x P --~ P is a Poisson map, where G x P is equipped with the product Poisson structure. Let g ~ G and x c P, denote by Crg : P --~ P and by ~rx : G -+ P the maps ere : x w-~ ~r(g, x) = gx, and o'x : g w+ o-(g, x) = gx, respectively, cr is called a Poisson action if
7re(gx) = % j r p ( x ) + Crx.rC~(g).
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Denote the infinitesimal action by ~. : g ~ F ( T P ) , equivalently )~(X)(x) = ax.X. For any f ~ C ~ ( P ) , let ~f E ~* be the covector defined by d
(~f(p), X) = -d~lt=of(etXp) = (df(p), X(X)).
(4.1)
If G is connected, o- is a Poisson action iff
X(X)({f, g}p) = {)~(X)(f), g}p + {f, X(X)(g)}p + (X, [~f, ~g].).
(4.2)
Clearly, if G has trivial Poisson structure, then G * = 9" has Lie Poisson structure and the G-action preserves the Poisson structure of P. Conversely, we have the following
PROPOSITION 4.1. Let a : G × P -+ P be a Poisson action. Poisson structure of P if and only if for every p ~ P, the isotropy subalgebra 9p at p is abelian. In particular, if the then a is an action preserving the Poisson structure if and Poisson Lie group.
Then it preserves the annihilator 9p of the action is locally free, only if G is a trivial
Proof: a preserves the Poisson structure of P if and only if )~(X)((f, gIp) = {X(X)(f), g)p + If, X(X)(g)}p.
(4.3)
Define the map A :d(C°°(P)) --+ 9% d f --+ ~f. Then by (4.1), the image of A is the annihilator of ~p = { X l ~ . ( X ) ( p ) = 0}. By (4.2) and (4.3), we know that the action preserves the Poisson structure of P if and only if for every p c P, the annihilator 9p of the isotropy subalgebra gp is abelian. If the action is locally flee, then the isotropy subalgebra 9p is trivial at any p 6 P, in order that the action preserves the Poisson structure, 9" must be an abelian Lie algebra, which means that the linearization of Poisson structure of G is zero, so the Poisson Lie structure of G is trivial. [] A Poisson action a : G x P --+ P is said to be tangential if every infinitesimal vector field X(X) is tangent to the symplectic leaf of P. A smooth map m : P -+ G* is called a momentum map if )~(X) = ~fip(m*Xt), here X 1 is the left invafiant 1-form on G whose value at the identity is X 6 9 = (9")*. A momentum map is called equivariant if it intertwines with the action of o- and the left dressing action of G on G*. A Poisson action with momentum map is clearly a tangential action. Dressing actions are tangential Poisson actions (cf. [13, 15]). The infinitesimal vector field of the left dressing action of G on G* is given by
dl (x) = 7r~G, (xl). Clearly by the definition of dressing action, the dressing orbits sweep out all symplectic leaves of G* and the identity map is an equivariant moment map. In the general case if the momentum map exists it is not necessarily equivariant [7]. Let m be a momentum map for the action a and define the map
Z : G x P ~ G*,
(g, x) w-~ (Dl(g)m(x)) -1. m(gx),
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101
then E measures the equivariance of m, it is equivariant if and only if E is a constant map. We want to give an infinitesimal description of E. Fix x c P, denote the following. E x : G --+ G*, g ~ E(g, x) and Fx, r(x) = (deEX(X), Y) for X, Y 6 g. Then we have PROPOSITION 4.2. (i)
F x, r = m * (zrG*( X l, yl) )
_ 7r p ( m , X 1,
m ,yl).
(4.4)
(ii)
7r~ (dF x, v) = 7C~p(m* ( i (d l (Y) )d X l) - i ( d ( X) )dY I)
(4.5)
+ i (X(X))d(m*Y l) - i (X(Y))d(m*Yl)). (iii) For Z c g, the differential of F x y in the direction of X(Z) is )~(Z)(Fx, r) = (L(Z), m*{X l, Y1}o, - {m*X L, m*yl}p)
+ d X 1 ( d ( Y ) - m,O~(Y)), m,(X(Z)))
(4.6)
+ d Y l ( ( m , X ( X ) ) - d l (X), m,()~(Z))). (iv) Define the map F " g x g -+ Coo(P), (X, Y) ~ and bilinear and
Fx, r. Then F is anti-symmetric
1 F , r ] ) ( x , Y, z ) ( d , r + ~[ = F([X, Y], Z) - (Z, [(m,(X(X))) l - (dl(X)) l, (m,(X(Y))) 1 - (d(Y))l],) + c.p.(X, Y, Z) = F([X, Y], Z) - (Lz(z)rce)(m*X l, m*Y l) + (Ld(z)Zr6,)(X t, yZ)
+ (LgZrG)(e)((dl(X)) l, (m,)~(Y)) t) + (LgZrG)(e)((m,X(X)) l, (dl(Y)) l) + c.p.(X, Y, Z),
(4.7)
here d • F ( A * T * P ) ~ F(A*+1T*P) is the usual de Rahm differential on forms defined by the Lie brackets of vector fields on P, and d," Horn(A'g, Coo(P))--> Hom(A*+IlL C°°(P)) is the de Rham differential defined by the Lie bracket of g which we denoted by [, ]; and Z is any vector field on G whose value at the unit of G is Z ~ t~. Proof: (i) E x maps the unit of G to that of G, so there is an induced map d E x : 1~ ~ g*. Using the definition of the tangent map, we have Fx, y(X) = (de~X(X), Y) = ((lm(x)-l),(m,(X(X))(x)) - (lm(x)_l),(dl(S)(m(x))), Y} = ( m , X ( X ) ( x ) - dz(X)(m(x)), yl)
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Q. YANG = (m,X(x),
-
o re(x),
= (m,(n.~(m,Xl)),
yl) _ (Tr~,(XI), yl) o m(x)
= m*(~G* (m(x))(X l, yl)) _ yrp (x)(m*X t, m*Yl).
(ii) From the formula (2.1), we have drx, r
= m*({x z, rl}6, - i(zrg, ( X t ) ) d r t + i(rc~, ( r l ) ) d X t ) -{m,X/, m*Yl}p + i (Zr~p(m* Xt) ) d m * Y l - i (rC~p(rn*Yl) )dm* X1 = m * { X l, YI}G, -
{m*X ¢, m * y l } p
+ i (Z (X))dm* yl _ i (k ( Y ) ) d m * X l - m* (i (d l ( X ) ) d Y l) + m* (i (d I (Y))dXl).
We use the following identity Zrp~(m*{X l, Y/}G*) -- Zr~p({m*Xl, m*Yl}p) = Zr~p(m,[X, y]l) _ [rr~p(m,Xl), rCp~(m,yl)] = 0 to cancel the first two terms in the expression of rr~p(dFx, y), then (4.5) follows. (iii) Note that Z(Z)(Fx,y) = (Z(Z), dFx, r), then from (4.5) we have i (Z (Z)) ((i (~. (X)))dm* yZ _ i (k ( Y ) ) d m * X I - m * (i ( d ( X ) ) d Y z) + m* (i (d l ( Y ) ) d X t )) ---dYt(m,()~(X)), m,(k(Z))) - d X Z ( m , ( k ( Y ) ) , m,(Z(Z))) - dYZ(dt(X), m,().(g))) + d X l ( d z ( Y ) , m,(Z(Z))) = d Y Z ( m , ( £ ( X ) ) - dl(X), m,(Z(Z))) + d X l ( d t ( Y ) - m , ( k ( Y ) ) , m,(k(Z))),
combining this with the expression of the dFz, y in the proof of (ii), we obtain
(4.6). (iv) Using the definition of the Schouten bracket of bivectors, we have
1 ~[F, FI(X, Y, Z) = (Z, F[X,
Y]r -
[FX, FY].),
(4.8)
and by definition (Z, F[X, r]r) = - ( F Z , ad)x Y - ad~r X) = -(Y, [FX, FZ],) - (X, [FZ, FY],). (4.9) By (4.4) (FX, Y) = (zr~,(xt), yZ) o m - (m,Or~(m*X~)), yl), (4.10) therefore we have F X = ( d ( X ) ) t - (m,(Z(X))) l.
By (4.8), (4.9) and (4.11) we have
(4.11)
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ALMOST HOMOGENEOUS POISSON MANIFOLDS
1 ( d , r + = [ r , r ] ) ( x , Y, z ) Z
= (Z, F[X, Y] - [rX, FY].) + c.p.(X, Y, Z)
= {rEx.y],z - (Z, [(m,(X(X))) l - (dl(X)) l, (m.(X(Y))) 1 - (dt(Y))l].)} + c.p.(X, Y, Z).
(4.12)
Since both G x P -+ P and G x G* ~ of [12],
G* are Poisson actions, by Theorem 4.7
(Lz(z~Zre)(m*X l, m*Y l) = (Z, [(m,(X(X))) l, (m, (X(Y)))I],), (Lal(z)Zrc,)(X t, yl) = (Z, [ ( d ( X ) ) 1, (dl (Y))l],).
(4.13) (4.14)
On the other hand, the Lie bracket [, ], is induced by the Poisson structure of G* which is defined by the Poisson structure of G. In fact we have
(LNTrG)(e)((dl(X)) l, (m.X(Y)) l) =. (Z, [(dl(X)) l, (m,),(Y))l],),
(4.15)
(Lg:,ra)(e)((m,X(X)) t, (dr(Y)) l) = (Z, [(m,Z(X)) z, (dz(Y))l],).
(4.16)
Substituting (4.13)-(4.16) into the last equality of (4.12) we obtain the last equality of (4.7). [] If G is connected, b y (i) of Proposition 4.2, the momentum map m : P --+ G* is equivariant if and only if m is a Poisson map. If P is a symplectic manifold, let x0 ~ P be such that m(xo) = e is the unit of G. Let A = m,zre(Xo) ~ A2g, and Jrtw = 7r~, + A r. Then Zrtw is still a Poisson structure of G. It is proved in [12] that m : (P, :re) ~ (G, Zrtw) is an equivariant momentum map. This means that m*(Jr~, + A r ) ( x l, yl)) _ Zre(m,X t, m,yZ) = 0. By (4.4) Fx, y(x) = Adm(x)-I A(X, Y). Since Zrtw is a Poisson structure, we have d A + ½[A, A] = 0. For a general Poisson manifold P, we conjecture that this equation holds on symplectic leaves of P. In the remaining part of this section we assume that G is a trivial Poisson Lie group. Then 7ra = 0 and G* = g*. Denote the set of Casimir functions on P by Car(P). In this case Fx, r ~ Car(P) is a Casimir function and
Fx, v = m([X, Y]) - {m(X), m(Y)}p, where we denote the X-component of m by re(X) 6 C°~(P). By definition the Poisson structure of P is preserved under the action of G, hence Lx(z)~p = Lax(z)Zra. = L~zra = 0, thus [F, F] = 0 and by (4.7)
d . r ( X , Y, Z) = {m([X, r ] ) , m(Z)}p -- m([[X, r ] , Z]) q- c.p.(X, Y, Z) = {{m(X), m(Y)]p - Fx, v, m(Z)}p - m([[X, Y], Z]) + c.p.(X, Y, Z) = {{re(X), rn(Y)}p, m(Z)}p - m([[X, Y], Z]) + c.p.(X, Y, Z) = 0.
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Therefore F 6 Hom(A2g, Car(P)) is a 2,cocycle relative to the trivial representation of g on Car(P)), or equivalently F E H2(g, Car(P)) ~ H 2 ( g ) ® Car(P). If HI(I~, C a r ( P ) ) = H2(g, C a r ( P ) ) = 0, in particular if g is semi-simple, the Poisson action o- always has a coadjoint equivariant momentum map. Let Fg,r(x) = (E(g, x), Y) be a function on P. If G is connected then Fg,r is also a Casimir function. Let L(~t, Car(P)) ~ g* @Car(P) be the set of linear functions from g to Car(P). Define a map • from G to L(g, Car(P)) by t p ( g ) ( y ) = Fg,r. Then ~ ( g h ) ( Y ) ( x ) = ~ ( g ) ( Y ) ( h x ) + ~(h)(Adg Y)(x). Since the action is tangential, hx and x lie in the same symplectic leaf and Fg,r is a Casimir function, we have ( ~ ( g ) ( Y ) ) ( h x ) = (Cb(g)(Y))(x). Hence t P ( g h ) = • (g) + A d g _ 1 ~ ( h ) and qJ is an L(g, Car(P))-valued l-cocycle on the group G, it represents a cohomology class in HI(G, Car(P)). Therefore G admits a coadjoint equivariant momentum map if and only if ~ is cohomologeous to zero. In particular, if G is compact, then it admits an equivariant momentum map.
5.
Almost homogeneous tangential Poisson actions
Let G be a Lie group and P a G-manifold. P is called G-almost homogeneous if G has at least one but at most finite number of open orbits in P. Note here that our definition of the almost homogeneous manifold is a little different from that of Huckleberry in [9], where the open orbit was unique, moreover where it was defined only for complex spaces. However, our definition is also applicable to the real case. If G is a Poisson Lie group and P is a G-almost homogeneous Poisson space, then P is an almost homogeneous Poisson G-manifold. If P is an almost homogeneous Poisson manifold under the tangential action of G, then each open G-orbit must be a symplectic leaf of P. The infinitesimal vector fields, which at the same time are Hamiltonian vector fields, generate the whole tangential space. Thus if an almost homogeneous Poisson G-action has an equivariant momentum map, then all open G-orbits in P are symplectic fibration spaces of the left dressing orbits in G. In particular, if P is homogeneous, P = O p = G . p = G/Gp for some p ~ P, then it is a symplectic fibration space of the G-dressing orbit O, = G/Gu, here u = re(p), and m : Op ~ (9, is a homogeneous fibration whose fibers are isomorphic to Gu/Gp. EXAMPLE 5.1. Let P be any connected compact holomorphic Poisson manifold. By Proposition 2.2, P is an almost homogeneous Ham(P)-manifold. Here the Poisson structure of P is preserved, however Ham(P) is in general not a finite-dimensional Lie group. PROPOSITION 5 . ] . Let ~r : G × P -+ P be a Poisson action on a symplectic manifold P admitting an equivariant momentum map m : P --+ G*. Then
Ker(m,p) = (TpOp) ~°,
Im(m,p) = (@)°,
(5.1)
ALMOST HOMOGENEOUS POISSON MANIFOLDS
105
here the upperscript co denotes the annihilator operation with respect to symplectic structure of P. Proof: V f ~ C ~ ( P ) , ( X ( X ) ( p ) , d f ) = (Tr~(m*Xl), d f ) = - ( m * X l, 7C~p(df)),
(5.2)
o)(p)O~(X), x f) = (U, x f).
(5.3)
thus Since P is symplectic, the tangential space TpP is spanned by the Hamilton vector fields, hence co(p)(X(X), v) = (X ~, re,v)), Vv E TpP. (5.4) Thus Ker(m,p) = {)~(X)(p)IX ~ 1~}~° = (TpOp) '° and Im(m,p) ° = { X t l X ( X ) ( p ) = 0} = {XIIX E t~p} = @, the later means Im(m,p) = (gtp)O. [] PROPOSITION 5.2. Let a : G x P --+ P be an almost homogeneous Poisson action on a symplectic manifold P admitting an equivariant momentum map m : P --+ G*, let Popen denote the union set of open G-orbits in P. Then Popen C Pmax and f o r any p C Popen, we have [Gu, Gu] C Gp, where u = m ( p ) .
Proof: Popen C Prn~x follows from the discussions at the beginning of this section, thus it suffices to prove that [Gu, Gu] C Gp. Let ~/ : [0, e] --~ Popen be a smooth curve with V(0) = p and g'(0) = V ( p ) c TpP. Note that dimgm(~,(t)) is locally constant near g(t), hence for any X, Y E gu, we can find smooth curves Xt, Yt E fJm(~'(t)). Therefore rcc,(m(v(t)))(Xlt, ytt ) = (zra,(Xt), ~ 1 Ylt)(m(g(t))) = (dl(X), g / ) ( m ( v ( t ) ) ) = O, hence yr~, (m (g (t)))(Xt, rt) = O.
(5.5)
Multiplying l(m(y(t)))-I with both sides of (5.5) then differentiating it at t = O, we get (L(m,V)ZCo*(X~, Y } ) = O. Using (2.2) and the fact that {X I, Y]}G* = [Xt, Yt] ~, we have (m, V, [Xt, rt] l) = "(7r~G,(Xl), d ( i ( m , V ) Y / ) ) + (rc~,(rl), d ( i ( m , V ) X l ) ) = 0.
(5.6)
Let V ( p ) run throughout the whole tangent space TpP, by (5.6) we must have [X, y]t c (Imm,) °. By (5.1) we immediately obtain [1~,, ~ ] C gp. [] THEOREM 5.1. Let G be a Poisson Lie group and P an almost homogeneous Poisson G-manifold with an equivariant momentum map m : P ~ G*. Then each open G-orbit in P is a symplectic fibre bundle over a dressing orbit. In particular, a compact homogeneous Poisson manifold with an equivariant momentum map is a symplectic torus bundle over a dressing orbit. Proof: The first part of this theorem follows from the discussions at the beginning of this section. If P is a compact homogeneous Poisson manifold with
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an equivariant momentum map and O, = G/G, C G*, here u bundle over the dressing orbit Gu/Gp and is compact as well.
m, let P = Op = G • p = G/Gp for some p E P, = m(p). Then P is a symplectic homogeneous fibre O~. Since P is compact, the fibre is isomorphic to By Proposition 5.2, P is a torus bundle over O,. []
Each dressing orbit is naturally a symplectic Poisson homogeneous manifold. It would be interesting to find more examples of tangential Poisson actions other than dressing actions. At the rest of this section we will give a family of examples of nontransitive almost homogeneous Poisson actions and give the conditions when they will be tangential Poisson actions. EXAMPLE 5.2. Consider the real special linear group
;) a,b,c,d~N, ad-bc=l].
SL(2, I R ) = [ ( ~ We denote el = ~
° o) 1
,
e2 = 2
1
:)
,
e3=~
1(: :)
sl(2, R). Then
is a basis of its Lie algebra [el, e2] = e3,
[e2, e3] = el,
[e3, el] = - e 2 .
Let A = )~1el A e2 -t- )~2e2 A e3 + )~3e3 A el, where )~1, )~2, )~3 c R. Clearly it is an r-matrix of sl(2, R) since A3s/(2, N[) is 1-dimensional and the elements of SL(2, N) have determinants 1. Let a - ( g ) = Lg, A - Rg.A with g ~ SL(2, IR). Then (SL(2, IR), zr) is a Poisson Lie group. The dual Lie bracket is defined by [~, 7/]. = ad~¢ t l - a d ~ ~ for any ~, ~ 6 s/*(2, R). Let e 1,* e 2,* e 3. be the basis of s/*(2, N) dual to el, e2, e3. It is easy to check that the Lie brackets for vectors of this basis are given by [e~, e~] = -)~3e~ - ~.2e~,
[e~, e~] = )~le~ - )~3e~,
[e 3, ell = )~2e~ - )~le~.
Let zrR2 = h(xl, x2)Oxl /x Ox2. Then (R 2, rr•2) be a Poisson manifold for any h(xl, x2) 6 C ~ ( R 2 ) . The natural action of SL(2, N) on IR2 is not transitive and has two orbits: a fixed point orbit O1 = {0} and an open orbit 0 2 = 11{2 - {0}. In order that the natural action ~r : SL(2, 1R) x (R 2, zra2) ~ (IR2, zra2) is a Poisson action, we must have rq~2(gx) = Crg,rC~2(x) + ~rx,Tr(g). (5.7) Let g =
(axa2)
E SL(2, IR),
a3
a4
x =
(xlt x2
ALMOST HOMOGENEOUS POISSON MANIFOLDS
107
Then (5.7) could be rewritten as
h(alxm + a2x2, a3xl +
a4x2) -
h(xi,
x2)
= 1{[(~.1 -~- ~.3)a 2 - - ()L1 - - )~3)a 2 - - 2 ) ~ 2 a l a 3 - - ()~1 + ~.3)] x 2
-'[- [(~.1 -1- ~.3)a2 -- (~q -- ~-3)a2 -- 2)~2a2a4 + (X1 -- ~.3)]x 2 + [2(;.1 + ~.3)ala2 -- 2(L1 -- ~.3)a3a4 --
4L2a2a3]xlx2}.
(5.8)
The solution of (5.8) is h(Xl, x2) =
1
~()~1 "-~ )~3) x 2 -
1
~ (~'1 - )~3)x 2 -
The infinitesimal action of the natural action of 1
(el)p
-m-
1 ~.2XlX2 + C,
SL(2, R)
1
~(XlOxl AX2Ox2),
(e2)p =
C e R.
(5.9)
on R 2 is 1 (e3)p = -~(XlOx 2 AX2Oxl),
~(XlOx2 AX2Oxl),
so this action is a tangential Poisson action if and only if ~.1, )~2, )~3, c satisfy +
> 0,
+
-
< 0,
c > 0.
(5.10)
For example, if ~.1 = ~2 -~- 0 and )~3 = 4, then h(xl, x 2 ) = x 2 + x 2 + c , the natural action a : SL(2, ~ ) × (I~2, rc~2) ~ (R 2, 7r~2) is a tangential Poisson action if c > 0. In the following, we consider the case )~1 = )~3 = 0 and ~'2 = 2 and take h(xl, x 2 ) = c- xlx2. In this case the dual Poisson Lie group SL*(2, ~ ) is realized explicitly by [12]:
SL,(2,~)defSB(2,]~)def{( ~ b+iC-1 ~a ] a > O,b,c ~ ] R , i 2 =
-1}
The dual basis is realized by
• (o
e1 =
,
,(::)
e2 =
and the dual Lie brackets are given by [e 1, e 2]
=
--2e 2,
[e2, e 3] = 0,
[e~, e~] = 2e~.
Note that (el) ° = Spana{e2, e3} is an ideal of sb(2, N), so H = Diag(a, a -1) is a Poisson Lie subgroup of SL(2, R) (cf. [18]). Each orbit of the restricted action of H on R 2 is of the form {(x, y) ~ R21xy = or}, where ot is any constant. Clearly this action preserves the Poisson structure and is tangential, it has a family of momentum maps:
m H : e 2 ---->~'~, ( X l ,
x 2 ) --+
f l 4 l c --
clearly they are H-equivariant momentum maps.
XlX2l ,
Vfl E I[~,
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Acknowledgements Part of this work was done while I was visiting as a guest fellow the Institut ftir Mathematik, Ruhr Universit~it Bochum, Germany. I would like to thank Prof. A. Huckleberry and E Heinzner for their financial supports. I also wish to thank Prof. Z.-J. Liu and Prof. J.-H. Lu for many useful comments. Finally, I would like to thank the referee for helpful comments and suggestions for this second version. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]
R. Abraham and J. E. Marsden: Foundation of Mechanics, 2nd edition, Benjamin 1978. A . A . Belaviu and V. G. Drinfeld: Math. Phys. Rev. 4 (1984), 93. Y. Brihaye, E. Kowalczyk and P. Maslanka: Acta Phys. Polon. 34 (2003), 2589. L . A . Cordero, M. Fernandez and R. Ib~i~ez and L. Ugarte: Ann. Glob. Ana. Geom. 18 (2000), 265. V.G. Drinfeld: Quantum groups, Proc. Cong. Math., Vol 1, Berkeley 1986. V.G. Drinfeld: Theor. Math. Phys. 95 (1993), 524. G. Ginzburg: Int. J. Math. 6 (1987), 330. A. Huckleberry and T. Wurzbacher: Math. Ann. 286 (1990), 261. A. Huckleberry and E. Oeljeklans: Classification theorems for almost homogeneous spaces, Institute Elie Cartan 1980. [10] S. Levendorskii and Y. Soibelman: Commun. Math. Phys. 139 (1991), 141. [11] Z.-J. Liu, A. Weinstein and P. Xu: J. Diff. Geom. 45 (1997), 547. [12] J.-H. Lu: Momentum mappings and reduction of Poisson action, S6minaire Sud-Rhodanien de G6om6trie 1989, MSRI publications, Springer, Berlin 1990. [13] J.-H. Lu and A. Weinstein: J. Diff. Geom. 31 (1990), 501. [14] K. Mikami: Trans. Amer. Math. Soc. 324 (1991), 447. [15] M . A . Semenov-Tian-Shansky: Publ. Res. Inst. Sci. Kyoto Univ. 21 (1985), 1237. [16] I. Vaisman: Lectures on the Geometry of Poisson Manifolds, 118, Birkh~iuser, Basel 1994. [17] A. Weinstein: J. Math. Fac. Sci. Univ. Tokyo 36, 163 (1988). [18] Q.-L. Yang: Adv. Math. Chin. 31 (2002), 127. [19] S. Zakrzewski: Commun. Math. Phys. 185 (1997), 285.
REPORTS ON MATHEMATICAL PHYSICS
No. 1
ALMOST HOMOGENEOUS POISSON MANIFOLDS QILIN
YANG
Mathematical Department, Tsinghua University, 100084, Beijing, P. R. China (e-mail: qlyang @math.tsinghua.edu.cn) (Received February 10, 2005 - Revised May 9, 2005)
We prove that any connected holomorphic Poisson manifold has an open and dense symplectic leaf which is a pseudo-Kiihler submanifold, and we define a new obstruction to study the equivariance of momentum map for tangential Poisson action. Some properties of almost homogeneous Poisson manifolds are studied and we prove that any compact symplectic Poisson homogeneous space is a toms bundle over a dressing orbit. Keywords: tangential Poisson action, almost homogeneous, momentum map. 2000 Mathematics Subject Classification: primary 53D17, 53D20; secondary 58F05, 58H05.
1.
Introduction
Motivated by the study of the Poisson nature of dressing transformation for soliton equations, Lu and Weinstein introduced the notion of Poisson action [12, 13, 17]. In the Poisson geometry a surprising fact is that Poisson action does not preserve Poisson structure in general, and it preserves the Poisson structure if and only if the Poisson Lie group has zero a Poisson structure. In this way Poisson actions are used to understand the "hidden symmetries" of certain integrable systems. Since Poisson manifold generalizes the concept of symplectic manifold, it is natural that the Poisson action generalizes the concept of symplectic action, and, in particular, the Hamiltonian action on symplectic manifold. Tangential Poisson action G × P --~ P is a special kind of Poisson action which is the most close generalization of Hamiltonian action. The orbits of tangential Poisson action locate in the symplectic leaves of P and the restricted action of G to every symplectic leaf is also a Poisson action, cf. [18]. Momentum mapping for a Hamiltonian action is the abstraction of momentum and angular momentum, which are associated with the translational and rotational invariance of the potential function of a classical mechanical system. The symmetry of the potential function leads to conservation of some quantities. This allows to simplify the description of mechanical system. In 1989, Lu generalized the concept of momentum map from Hamiltonian action to tangential Poisson action and studied the Meyer-Marsden-Weinstein momentum reduction for tangential Poisson action on symplectic manifold in regular case [12]. Later on, Ginzburg investigated systematically the existence and uniqueness of [93]
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Q. YANG
equivariant momentum map for tangential Poisson action in [7] via calculating the Poisson cohomology. Recently we generalized the momentum 'reduction to singular value level for general tangential Poisson actions in [18]. In this paper, we discuss other properties of the momentum mapping for tangential Poisson actions. The paper is organized as follows: in Section 2, we give a detailed description of symplectic stratification of a Poisson manifold and emphasize the symplectic stratification of a holomorphic Poisson manifold. In Section 3, we survey the known dispersed results on the classifications of Poisson Lie groups and investigate Poisson Lie group structures on Abelian Lie groups which have been not considered so far. In Section 4, we give the necessary and sufficient conditions for an action to preserve the Poisson structure, and define an obstruction in describing the equivariance of momentum map for a general Poisson action which does not necessarily preserves the Poisson structure. As far as I know, all known until now obstructions were studied for some special kinds of Poisson actions when the Poisson structures were invariant under such actions. In Section 5, we prove that the compact symplectic Poisson homogeneous space is a torus bundle over some dressing orbit, generalizing similar results for Hamiltonian actions given in [8]. At the end, for the first time, we give an example of nontransitive tangential Poisson action.
2.
Holomorphic Poisson manifolds and their symplectic leaves A (real) n-dimensional differentiable manifold P is called a Poisson manifold if the set of smooth functions C ~ ( P ) is a Poisson algebra over real number field R. If
we denote the Poisson bracket of functions f and g by {f, g}p, then the equivalent definition is that {f, .}p is a derivation on the commutative algebra C ~ ( P ) and meanwhile ( C ~ ( P ) , {, }~) is a Lie algebra over JR. We can conclude (cf. [16]) that there exists a bivector Jrp c F(A2Tp), called Poisson bivector field, such that the Schouten bracket [:rp, 7rp] = 0, which just corresponds to the Jacobi identity for Poisson bracket. Let ~z~ • F(T*P) --~ F ( T P ) be the bundle map defined by (:r~(/~), a) = zrp(a, I3). Then X I := -Tr~(df) : { , f } is a vector field, called the Hamiltonian vector field associated with function f. The Poisson structure induces a Lie algebroid structure (cf. [11]) on the cotangent bundle T*P. The Lie bracket on the space F ( T * P ) of 1-forms, still denoted by {, }p, is given by (cf. [12, 16]): {oe, 15}p = d(a'p(o~,/3) - (;v~ (oe), d]5) + (n'~ (/3), doO,
(2.1)
if V E I'.(TP) is a vector field, then (V, {a, fl}p) = (Lv:rp)(a, t ) - (:rr~(a), d(i(V)fl)) + (:'r~(fl), d(i(V)a)).
(2.2)
For a Poisson manifold P which is a complex manifold, clearly the algebra of complex-valued smooth functions C ~ ( P , C), coinciding with the base extended algebra C ~ ( P ) ® ~ C, is in a natural way a complex Poisson algebra. This fact tell us that studying complex Poisson algebra of smooth functions is to some extent
ALMOST HOMOGENEOUS POISSON MANIFOLDS
95
reduced to the case of real Poisson algebra of smooth functions. In other words, the classifications of smooth Poisson structures on real and complex manifolds have no essential differences. To stress the impact of the complex structure on the Poisson structure :re, we introduce the following definition. DEFINITION 2.1. A complex manifold P with Poisson bivector field :rp c F(A2TI'°P) is called a holomorphic Poisson manifold, if :rp is also a holomorphic bivector field, where TI,°P denotes the holomorphic tangential bundle of P. Similarly to the real case, it is very easy to verify that a complex manifold P is a holomorphic Poisson manifold iff the algebra O(P) of holomorphic functions is a complex Poisson algebra. We remark here that our definition of holomorphic Poisson manifold coincides with that in [4, Definition 3.1], where it is called holomorphic complex Poisson structure. However, in [4] the authors investigated mainly almost complex Poisson manifold, which means that the underlying manifold is only an almost complex manifold. Hence what they called almost complex Poisson manifolds are more general objects than holomorphic Poisson manifolds considered here. For any Poisson manifold P of real dimension n, if the Poisson bivector :re is a nondegenerate tensor, the inverse bundle map of :r~ thought of as a two-form is a symplectic form [16], therefore P itself is the unique symplectic leaf. However, in general case, :re may have varying rank. Let
D = {V • r ( T P ) I 3 f e C°°(P), X f ( p ) = V(p), Vp E P}.. D is a differential distribution on P. By Jacobi identity we have [X f, Xs] = X{I,~}. So D is an involutive distribution. In fact, it is a completely integrable distribution (cf. [16, Theorem 2.6]). We denote the integral leaf through p by .Tp. Since the Hamiltonian flows defined by Hamiltonian vector fields preserve the Poisson brackets, the restriction of :rp to .Tp has constant rank. In this way we know that ,Tp is a symplectic submanifold of P and hence a symplectic leaf of P. If P is compact, let Ham(P) C Diff(P) be the Hamiltonian diffeomorphism group with Lie algebra { X f l f • C ~ ( P ) } . Then the symplectic leaves are just the connected components of Ham(P)-orbits in P. Let r(p) := dimRffp, then r(p) is equal to the rank of :rp at p. Since Jr/, is skew symmetric, r is an even integer-valued function bounded by the dimension of P. Naturally, P is decomposed into the union of symplectic leaves with varying dimensions,
U
peP O
If moreover P is a holomorphic Poisson manifold, the above decomposition has subtle structure. Let m denote the complex dimension of P and thus the real dimension of P is n = 2m. Let (Zl, z2 . . . . , Zm) be a local holomorphic coordinate system at p • P, and write :re(P) = Ei,j :retj 8zi /X 8zj. For a positive integer k < m and two k-tuples Ik = (il, i2 . . . . . ik) and Jk = (jl, j2 . . . . . jk), let :r/k& be the
96
Q. YANG
ij " " submatrix of (rOe)toxin, whose entry at itx-th row and jv-th column is ZrpuJ~, here 1 < /z, v < k. Let det(zr~~&) denote the determinant of the k-order square matrix Ik4 Jrf , and define the functions
dk = ~ Ldet(@k4)12, Ik,J~
k = 1,2 . . . . . m,
ij where summation runs on all k x k submatrices of (Zrp)mxm. We still use r(p) to denote the real dimension of a symplectic leaf passing through p e P, and denote 'max' the maximum of r. Let PI = {P e P I r(p) = l} and QI = {p e P I r(p) < l}. Then Pmax is the union of symplectic leaves of P where the Poisson structure has maximal rank and m
P = U P21 = Qmax. k=l
PROPOSITION 2.1. I f P is a holomorphic Poisson manifold, then for any positive integer l, the set P2l-1 is empty; if Q2l is nonempty, then the real codimension of the subset Q2t-2 in Q2l is at least 2.
Proof: Clearly P2l-1 is empty since the rank of a skew symmetric matrix is always even. We have l
Q21 = U P2k = {P E P I dl+l(p) = dl+2(p) . . . . .
din(p) = 0},
k=l
where the last equality follows from the definition of the rank of a matrix. Therefore we have Q2l-2 = {p E Q21 I d l ( p ) = 0}. Since locally all determinants of submatrices det(zrfkJk) are holomorphic functions on P for any k-tuple Ik and Jk, all Q2~ are analytic subsets of P. If Q21 is nonempty, then there exists at least one nondegenerate submatrix Jr) J~ of l-th order. Note that
Q2l-2 = {P ¢ Q21 I dr(p) = O} C {p E Q21 I det(rcY l) = O} and d e t U r y z) is a nonzero holomorphic function, thus the real codimension of Q21-2 in Q21 is at least 2. [] COROLLARY 2.1. For a holomorphic Poisson manifold P, if P2z is nonempty, then P2z is a relative open subset in Q21. In particular, Pmax is an open connected density subset of P if P is a connected holomorphic Poisson manifold.
Proof: From the proof of Proposition 2.1 we know that Q2z = P2l u Q2l-2. Since Q21 and Q2z-2 are closed subsets, by Proposition 2.1, P2I is a relative open subset in Q2z. Moreover Pmax is the complement of the closed subset Qmax-2 in Qma~ = P, and Qm~x-2 is a subset in Qmax of a real codimension equal at least to two. Note
ALMOST HOMOGENEOUS POISSON MANIFOLDS
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that any subset of real codimension two does not destroy the connectedness of a manifold, so Pmax is an open connected dense subset of P. [] Any smooth Poisson manifold P of real dimension n has the rank decomposition P = U0_
U
)~A k, l
f;"
The symplectic leaves in Pmax are all open. So, the Ham(P)-action has at least one open orbit. If (P, :re, J) is a holomorphic Poisson manifold with complex structure J, the Hamiltonian vector field of any holomorphic function is a holomorphic vector field. So all symplectic leaves are complex symplectic submanifolds and the restriction of rrp to z the symplectic leaf Y:~ defines a symplectic form wkz satisfying cok(J., J.) = cokz (., .). Therefore all symplectic leaves 5c2 are pseudo-K~ihler submanifolds of P. Thus we have the following result. PROPOSITION 2.2. Let P be a holomorphic Poisson manifold of complex dimension m. Then we have a symplectic decomposition
t,=
U
~.~A k, 1 < k < m
where all symplectic leaves ~ are pseudo-Kiihler submanifolds of P. Moreover, P has at least one open symplectic leaf which is dense in P if P is connected. 3.
Poisson Lie groups
A smooth map between smooth Poisson manifolds M and N is called a Poisson map if the pull-back map f * : C ~ ( N ) -~ C ~ ( M ) is a Poisson algebra homomorphism. Given two Poisson manifolds (M, r~M) and (N, rCN), there is a unique Poisson structure, denoted by JrM G row, on the product manifold M × N, such that the projections from M × N to each factor M and N are both Poisson maps. We called it the product Poisson structure. A Lie group which is at the same
98
Q. YANG
time a Poisson manifold is called a Poisson Lie group if the multiplicative map lz : G x G --+ G, (g, h) --+ gh is a Poisson map, where G × G is equipped with the product Poisson structure. Let rg and lg respectively denote the left and right translation in G by g e G. Then the Poisson bivector field zrG satisfies rcG(gh) = Igzr~(g) + rhZrG(g),
Vg, h c G,
(3.1)
and is called a multiplicative bivector field. If b o t h g, h are specialized to be the unit e of G, then zrG(e)= 0. So a Poisson Lie group can never be a symplectic manifold. Let 3r~(g) = rgrCG(g). Then rc~(gh) = 7r~(g) + Adg zr~(h),
Vg, h e G,
so the linearization of zr~ at the unit is a A2~[ -valued 1-cocycle on g = Z;ie(G) relative to the adjoint action of g on A21~, that is 6 = dezr~ e H 1(~1, A2g). If 3 is a 1-coboundary, then G is called a coboundary Poisson-Lie group. The dual map 6* defines a Lie algebra structure on the dual space g* of g. There is a Lie algebra structure on g @ g* such that both g and g* are Lie subalgebras. (g ® g*, g, ~t*) is called the Lie bialgebra of (G, zrG). There is a 1-1 correspondence between connected and simply connected Poisson Lie groups and Lie bialgebras (cf. [13]). The connected and simply connected Lie group G* with Lie algebra g* is also a Poisson Lie group, called the dual Poisson Lie group of G. A Lie group D ( G ) with Lie algebra ~I @ g* is called a double for the Poisson Lie group G if the multiplication map G x G C D ( G ) , (g, h) ~ gh is a diffeomorphism. In this case, the left-action of D ( G ) on itself induces an action on G*, restricting this action to subgroup G we obtain an action G x G* --+ G*, (g, h) ~-> D l (g), it is called the left dressing action of G on G*. If G has trivial Poisson Lie group structure, then G* = g* has Lie-Poisson structure, and the left dressing action is just the coadjoint action of G on g*. The definition of right dressing action is likewise (cf. [12]). If G is semisimple, the Whitehead theorem says that both Hi(g, V) and H2(I~, V) vanish for any ~t-module V. Thus every connected semi-simple Poisson Lie group is coboundary Poisson Lie group. The 1-cocycle is given by 3 (X) = adx A for some _A E A2t~. By integration we get a 2-cycle ZrG(g) = lgA - r g A on the group G. Clearly zr6 is a Poisson bivector if and only if [A, A] e A3g is Adc-invariant. Such a A is called an r-matrix. Hence all Poisson Lie group structures on semisimple Lie groups are defined by r-matrices. Let G = T m x R n be a direct product of m-dimensional torus group and n-dimensional vector group IRn and let zr~(01 . . . . . Om, xl . . . . . x , ) =
--
G~(eiO1 . . . . . e i°m,xl . . . . . Xn)
7"g r
(3.2)
be the lift of 7r~ on the universal covering space N m+n. By the multiplicative condition (3.1), we know that (~rc)i j is a linear function with respect to the variables 0k and xl and periodic with respect to Ok. So the Poisson bivector has
ALMOST HOMOGENEOUS POISSON MANIFOLDS
the form
99
m-}-n
: } 2 C/ u 0.i A 0.j,
(3.3)
i,j,k=l
where the C/~ are the structure constants of an (m + n)-dimensional Lie algebra and C~. = 0 for k < m; ui =Oi for 1 < i < m and Um+i = x i for 1 < i < n. In particular, if G is the torus T m, is has only zero Poisson Lie group, thus any Poisson Lie group structure on compact abelian group is trivial. If G is the vector group IRn, the nontrivial Poisson Lie group structure on it must be linear and is called a Lie-Poisson structure. For a multiplicative abelian group G = (R+) n or (C*)", the solution of (3.1) is of the form n
(zr;)u~(zl . . . . . zn) = ~
C ~ lnz~,
(3.4)
k=l
rrc is a Poisson structure iff C ~ = - C ~ . for 1 < / x , v, 8 < n and n Z
/z
v
/x
v
p,
v
[(Cp~ C8× + Cp~,C~×) In zu In z~ + Cp, C~y In zu] + c.p. (p, 8, Y) = 0,
(3.5)
/z,v=l
where 1 < p, 8, F --- n, and c.p.(p, 8, F) means cyclic permutation with respect to p, 8 and g. After some simple calculations, we know that (3.5) is equivalent to n Z
/z v /x v /z v (Cp~C~, + C~,C×p + Cy~Cp~) = 0,
1 < Ix, p, 8, g <- n.
(3.6)
v=l
So Cp~ are also the structure constants of an n-dimensional Lie algebra. Therefore the Poisson Lie group structures on noncompact abelian groups are essentially equivalent to the Lie algebra structures. Belavin and Drinfeld classified all Poisson Lie group structures on complex simple Lie groups [2], Levendorskii and Soibelman obtained the classification of Poisson-Lie' group structures on compact connected Lie groups in [10]. The classification of Poisson-Lie group structures on a non-reductive Lie group is more involved. The cases of 2-dimensional affine group, Poincar6 group and Galilei group were solved completely in [3, 14, 19].
4.
Poisson actions and momentum mappings
An action of a Poisson Lie group (G, zrG) on a Poisson manifold (P, rre) is called a Poisson action (cf. [13, 15]) if the action map ~r : G x P --~ P is a Poisson map, where G x P is equipped with the product Poisson structure. Let g ~ G and x c P, denote by Crg : P --~ P and by ~rx : G -+ P the maps ere : x w-~ ~r(g, x) = gx, and o'x : g w+ o-(g, x) = gx, respectively, cr is called a Poisson action if
7re(gx) = % j r p ( x ) + Crx.rC~(g).
100
Q. YANG
Denote the infinitesimal action by ~. : g ~ F ( T P ) , equivalently )~(X)(x) = ax.X. For any f ~ C ~ ( P ) , let ~f E ~* be the covector defined by d
(~f(p), X) = -d~lt=of(etXp) = (df(p), X(X)).
(4.1)
If G is connected, o- is a Poisson action iff
X(X)({f, g}p) = {)~(X)(f), g}p + {f, X(X)(g)}p + (X, [~f, ~g].).
(4.2)
Clearly, if G has trivial Poisson structure, then G * = 9" has Lie Poisson structure and the G-action preserves the Poisson structure of P. Conversely, we have the following
PROPOSITION 4.1. Let a : G × P -+ P be a Poisson action. Poisson structure of P if and only if for every p ~ P, the isotropy subalgebra 9p at p is abelian. In particular, if the then a is an action preserving the Poisson structure if and Poisson Lie group.
Then it preserves the annihilator 9p of the action is locally free, only if G is a trivial
Proof: a preserves the Poisson structure of P if and only if )~(X)((f, gIp) = {X(X)(f), g)p + If, X(X)(g)}p.
(4.3)
Define the map A :d(C°°(P)) --+ 9% d f --+ ~f. Then by (4.1), the image of A is the annihilator of ~p = { X l ~ . ( X ) ( p ) = 0}. By (4.2) and (4.3), we know that the action preserves the Poisson structure of P if and only if for every p c P, the annihilator 9p of the isotropy subalgebra gp is abelian. If the action is locally flee, then the isotropy subalgebra 9p is trivial at any p 6 P, in order that the action preserves the Poisson structure, 9" must be an abelian Lie algebra, which means that the linearization of Poisson structure of G is zero, so the Poisson Lie structure of G is trivial. [] A Poisson action a : G x P --+ P is said to be tangential if every infinitesimal vector field X(X) is tangent to the symplectic leaf of P. A smooth map m : P -+ G* is called a momentum map if )~(X) = ~fip(m*Xt), here X 1 is the left invafiant 1-form on G whose value at the identity is X 6 9 = (9")*. A momentum map is called equivariant if it intertwines with the action of o- and the left dressing action of G on G*. A Poisson action with momentum map is clearly a tangential action. Dressing actions are tangential Poisson actions (cf. [13, 15]). The infinitesimal vector field of the left dressing action of G on G* is given by
dl (x) = 7r~G, (xl). Clearly by the definition of dressing action, the dressing orbits sweep out all symplectic leaves of G* and the identity map is an equivariant moment map. In the general case if the momentum map exists it is not necessarily equivariant [7]. Let m be a momentum map for the action a and define the map
Z : G x P ~ G*,
(g, x) w-~ (Dl(g)m(x)) -1. m(gx),
ALMOST HOMOGENEOUS POISSON MANIFOLDS
101
then E measures the equivariance of m, it is equivariant if and only if E is a constant map. We want to give an infinitesimal description of E. Fix x c P, denote the following. E x : G --+ G*, g ~ E(g, x) and Fx, r(x) = (deEX(X), Y) for X, Y 6 g. Then we have PROPOSITION 4.2. (i)
F x, r = m * (zrG*( X l, yl) )
_ 7r p ( m , X 1,
m ,yl).
(4.4)
(ii)
7r~ (dF x, v) = 7C~p(m* ( i (d l (Y) )d X l) - i ( d ( X) )dY I)
(4.5)
+ i (X(X))d(m*Y l) - i (X(Y))d(m*Yl)). (iii) For Z c g, the differential of F x y in the direction of X(Z) is )~(Z)(Fx, r) = (L(Z), m*{X l, Y1}o, - {m*X L, m*yl}p)
+ d X 1 ( d ( Y ) - m,O~(Y)), m,(X(Z)))
(4.6)
+ d Y l ( ( m , X ( X ) ) - d l (X), m,()~(Z))). (iv) Define the map F " g x g -+ Coo(P), (X, Y) ~ and bilinear and
Fx, r. Then F is anti-symmetric
1 F , r ] ) ( x , Y, z ) ( d , r + ~[ = F([X, Y], Z) - (Z, [(m,(X(X))) l - (dl(X)) l, (m,(X(Y))) 1 - (d(Y))l],) + c.p.(X, Y, Z) = F([X, Y], Z) - (Lz(z)rce)(m*X l, m*Y l) + (Ld(z)Zr6,)(X t, yZ)
+ (LgZrG)(e)((dl(X)) l, (m,)~(Y)) t) + (LgZrG)(e)((m,X(X)) l, (dl(Y)) l) + c.p.(X, Y, Z),
(4.7)
here d • F ( A * T * P ) ~ F(A*+1T*P) is the usual de Rahm differential on forms defined by the Lie brackets of vector fields on P, and d," Horn(A'g, Coo(P))--> Hom(A*+IlL C°°(P)) is the de Rham differential defined by the Lie bracket of g which we denoted by [, ]; and Z is any vector field on G whose value at the unit of G is Z ~ t~. Proof: (i) E x maps the unit of G to that of G, so there is an induced map d E x : 1~ ~ g*. Using the definition of the tangent map, we have Fx, y(X) = (de~X(X), Y) = ((lm(x)-l),(m,(X(X))(x)) - (lm(x)_l),(dl(S)(m(x))), Y} = ( m , X ( X ) ( x ) - dz(X)(m(x)), yl)
102
Q. YANG = (m,X(x),
-
o re(x),
= (m,(n.~(m,Xl)),
yl) _ (Tr~,(XI), yl) o m(x)
= m*(~G* (m(x))(X l, yl)) _ yrp (x)(m*X t, m*Yl).
(ii) From the formula (2.1), we have drx, r
= m*({x z, rl}6, - i(zrg, ( X t ) ) d r t + i(rc~, ( r l ) ) d X t ) -{m,X/, m*Yl}p + i (Zr~p(m* Xt) ) d m * Y l - i (rC~p(rn*Yl) )dm* X1 = m * { X l, YI}G, -
{m*X ¢, m * y l } p
+ i (Z (X))dm* yl _ i (k ( Y ) ) d m * X l - m* (i (d l ( X ) ) d Y l) + m* (i (d I (Y))dXl).
We use the following identity Zrp~(m*{X l, Y/}G*) -- Zr~p({m*Xl, m*Yl}p) = Zr~p(m,[X, y]l) _ [rr~p(m,Xl), rCp~(m,yl)] = 0 to cancel the first two terms in the expression of rr~p(dFx, y), then (4.5) follows. (iii) Note that Z(Z)(Fx,y) = (Z(Z), dFx, r), then from (4.5) we have i (Z (Z)) ((i (~. (X)))dm* yZ _ i (k ( Y ) ) d m * X I - m * (i ( d ( X ) ) d Y z) + m* (i (d l ( Y ) ) d X t )) ---dYt(m,()~(X)), m,(k(Z))) - d X Z ( m , ( k ( Y ) ) , m,(Z(Z))) - dYZ(dt(X), m,().(g))) + d X l ( d z ( Y ) , m,(Z(Z))) = d Y Z ( m , ( £ ( X ) ) - dl(X), m,(Z(Z))) + d X l ( d t ( Y ) - m , ( k ( Y ) ) , m,(k(Z))),
combining this with the expression of the dFz, y in the proof of (ii), we obtain
(4.6). (iv) Using the definition of the Schouten bracket of bivectors, we have
1 ~[F, FI(X, Y, Z) = (Z, F[X,
Y]r -
[FX, FY].),
(4.8)
and by definition (Z, F[X, r]r) = - ( F Z , ad)x Y - ad~r X) = -(Y, [FX, FZ],) - (X, [FZ, FY],). (4.9) By (4.4) (FX, Y) = (zr~,(xt), yZ) o m - (m,Or~(m*X~)), yl), (4.10) therefore we have F X = ( d ( X ) ) t - (m,(Z(X))) l.
By (4.8), (4.9) and (4.11) we have
(4.11)
103
ALMOST HOMOGENEOUS POISSON MANIFOLDS
1 ( d , r + = [ r , r ] ) ( x , Y, z ) Z
= (Z, F[X, Y] - [rX, FY].) + c.p.(X, Y, Z)
= {rEx.y],z - (Z, [(m,(X(X))) l - (dl(X)) l, (m.(X(Y))) 1 - (dt(Y))l].)} + c.p.(X, Y, Z).
(4.12)
Since both G x P -+ P and G x G* ~ of [12],
G* are Poisson actions, by Theorem 4.7
(Lz(z~Zre)(m*X l, m*Y l) = (Z, [(m,(X(X))) l, (m, (X(Y)))I],), (Lal(z)Zrc,)(X t, yl) = (Z, [ ( d ( X ) ) 1, (dl (Y))l],).
(4.13) (4.14)
On the other hand, the Lie bracket [, ], is induced by the Poisson structure of G* which is defined by the Poisson structure of G. In fact we have
(LNTrG)(e)((dl(X)) l, (m.X(Y)) l) =. (Z, [(dl(X)) l, (m,),(Y))l],),
(4.15)
(Lg:,ra)(e)((m,X(X)) t, (dr(Y)) l) = (Z, [(m,Z(X)) z, (dz(Y))l],).
(4.16)
Substituting (4.13)-(4.16) into the last equality of (4.12) we obtain the last equality of (4.7). [] If G is connected, b y (i) of Proposition 4.2, the momentum map m : P --+ G* is equivariant if and only if m is a Poisson map. If P is a symplectic manifold, let x0 ~ P be such that m(xo) = e is the unit of G. Let A = m,zre(Xo) ~ A2g, and Jrtw = 7r~, + A r. Then Zrtw is still a Poisson structure of G. It is proved in [12] that m : (P, :re) ~ (G, Zrtw) is an equivariant momentum map. This means that m*(Jr~, + A r ) ( x l, yl)) _ Zre(m,X t, m,yZ) = 0. By (4.4) Fx, y(x) = Adm(x)-I A(X, Y). Since Zrtw is a Poisson structure, we have d A + ½[A, A] = 0. For a general Poisson manifold P, we conjecture that this equation holds on symplectic leaves of P. In the remaining part of this section we assume that G is a trivial Poisson Lie group. Then 7ra = 0 and G* = g*. Denote the set of Casimir functions on P by Car(P). In this case Fx, r ~ Car(P) is a Casimir function and
Fx, v = m([X, Y]) - {m(X), m(Y)}p, where we denote the X-component of m by re(X) 6 C°~(P). By definition the Poisson structure of P is preserved under the action of G, hence Lx(z)~p = Lax(z)Zra. = L~zra = 0, thus [F, F] = 0 and by (4.7)
d . r ( X , Y, Z) = {m([X, r ] ) , m(Z)}p -- m([[X, r ] , Z]) q- c.p.(X, Y, Z) = {{m(X), m(Y)]p - Fx, v, m(Z)}p - m([[X, Y], Z]) + c.p.(X, Y, Z) = {{re(X), rn(Y)}p, m(Z)}p - m([[X, Y], Z]) + c.p.(X, Y, Z) = 0.
104
Q. YANG
Therefore F 6 Hom(A2g, Car(P)) is a 2,cocycle relative to the trivial representation of g on Car(P)), or equivalently F E H2(g, Car(P)) ~ H 2 ( g ) ® Car(P). If HI(I~, C a r ( P ) ) = H2(g, C a r ( P ) ) = 0, in particular if g is semi-simple, the Poisson action o- always has a coadjoint equivariant momentum map. Let Fg,r(x) = (E(g, x), Y) be a function on P. If G is connected then Fg,r is also a Casimir function. Let L(~t, Car(P)) ~ g* @Car(P) be the set of linear functions from g to Car(P). Define a map • from G to L(g, Car(P)) by t p ( g ) ( y ) = Fg,r. Then ~ ( g h ) ( Y ) ( x ) = ~ ( g ) ( Y ) ( h x ) + ~(h)(Adg Y)(x). Since the action is tangential, hx and x lie in the same symplectic leaf and Fg,r is a Casimir function, we have ( ~ ( g ) ( Y ) ) ( h x ) = (Cb(g)(Y))(x). Hence t P ( g h ) = • (g) + A d g _ 1 ~ ( h ) and qJ is an L(g, Car(P))-valued l-cocycle on the group G, it represents a cohomology class in HI(G, Car(P)). Therefore G admits a coadjoint equivariant momentum map if and only if ~ is cohomologeous to zero. In particular, if G is compact, then it admits an equivariant momentum map.
5.
Almost homogeneous tangential Poisson actions
Let G be a Lie group and P a G-manifold. P is called G-almost homogeneous if G has at least one but at most finite number of open orbits in P. Note here that our definition of the almost homogeneous manifold is a little different from that of Huckleberry in [9], where the open orbit was unique, moreover where it was defined only for complex spaces. However, our definition is also applicable to the real case. If G is a Poisson Lie group and P is a G-almost homogeneous Poisson space, then P is an almost homogeneous Poisson G-manifold. If P is an almost homogeneous Poisson manifold under the tangential action of G, then each open G-orbit must be a symplectic leaf of P. The infinitesimal vector fields, which at the same time are Hamiltonian vector fields, generate the whole tangential space. Thus if an almost homogeneous Poisson G-action has an equivariant momentum map, then all open G-orbits in P are symplectic fibration spaces of the left dressing orbits in G. In particular, if P is homogeneous, P = O p = G . p = G/Gp for some p ~ P, then it is a symplectic fibration space of the G-dressing orbit O, = G/Gu, here u = re(p), and m : Op ~ (9, is a homogeneous fibration whose fibers are isomorphic to Gu/Gp. EXAMPLE 5.1. Let P be any connected compact holomorphic Poisson manifold. By Proposition 2.2, P is an almost homogeneous Ham(P)-manifold. Here the Poisson structure of P is preserved, however Ham(P) is in general not a finite-dimensional Lie group. PROPOSITION 5 . ] . Let ~r : G × P -+ P be a Poisson action on a symplectic manifold P admitting an equivariant momentum map m : P --+ G*. Then
Ker(m,p) = (TpOp) ~°,
Im(m,p) = (@)°,
(5.1)
ALMOST HOMOGENEOUS POISSON MANIFOLDS
105
here the upperscript co denotes the annihilator operation with respect to symplectic structure of P. Proof: V f ~ C ~ ( P ) , ( X ( X ) ( p ) , d f ) = (Tr~(m*Xl), d f ) = - ( m * X l, 7C~p(df)),
(5.2)
o)(p)O~(X), x f) = (U, x f).
(5.3)
thus Since P is symplectic, the tangential space TpP is spanned by the Hamilton vector fields, hence co(p)(X(X), v) = (X ~, re,v)), Vv E TpP. (5.4) Thus Ker(m,p) = {)~(X)(p)IX ~ 1~}~° = (TpOp) '° and Im(m,p) ° = { X t l X ( X ) ( p ) = 0} = {XIIX E t~p} = @, the later means Im(m,p) = (gtp)O. [] PROPOSITION 5.2. Let a : G x P --+ P be an almost homogeneous Poisson action on a symplectic manifold P admitting an equivariant momentum map m : P --+ G*, let Popen denote the union set of open G-orbits in P. Then Popen C Pmax and f o r any p C Popen, we have [Gu, Gu] C Gp, where u = m ( p ) .
Proof: Popen C Prn~x follows from the discussions at the beginning of this section, thus it suffices to prove that [Gu, Gu] C Gp. Let ~/ : [0, e] --~ Popen be a smooth curve with V(0) = p and g'(0) = V ( p ) c TpP. Note that dimgm(~,(t)) is locally constant near g(t), hence for any X, Y E gu, we can find smooth curves Xt, Yt E fJm(~'(t)). Therefore rcc,(m(v(t)))(Xlt, ytt ) = (zra,(Xt), ~ 1 Ylt)(m(g(t))) = (dl(X), g / ) ( m ( v ( t ) ) ) = O, hence yr~, (m (g (t)))(Xt, rt) = O.
(5.5)
Multiplying l(m(y(t)))-I with both sides of (5.5) then differentiating it at t = O, we get (L(m,V)ZCo*(X~, Y } ) = O. Using (2.2) and the fact that {X I, Y]}G* = [Xt, Yt] ~, we have (m, V, [Xt, rt] l) = "(7r~G,(Xl), d ( i ( m , V ) Y / ) ) + (rc~,(rl), d ( i ( m , V ) X l ) ) = 0.
(5.6)
Let V ( p ) run throughout the whole tangent space TpP, by (5.6) we must have [X, y]t c (Imm,) °. By (5.1) we immediately obtain [1~,, ~ ] C gp. [] THEOREM 5.1. Let G be a Poisson Lie group and P an almost homogeneous Poisson G-manifold with an equivariant momentum map m : P ~ G*. Then each open G-orbit in P is a symplectic fibre bundle over a dressing orbit. In particular, a compact homogeneous Poisson manifold with an equivariant momentum map is a symplectic torus bundle over a dressing orbit. Proof: The first part of this theorem follows from the discussions at the beginning of this section. If P is a compact homogeneous Poisson manifold with
106
Q. YANG
an equivariant momentum map and O, = G/G, C G*, here u bundle over the dressing orbit Gu/Gp and is compact as well.
m, let P = Op = G • p = G/Gp for some p E P, = m(p). Then P is a symplectic homogeneous fibre O~. Since P is compact, the fibre is isomorphic to By Proposition 5.2, P is a torus bundle over O,. []
Each dressing orbit is naturally a symplectic Poisson homogeneous manifold. It would be interesting to find more examples of tangential Poisson actions other than dressing actions. At the rest of this section we will give a family of examples of nontransitive almost homogeneous Poisson actions and give the conditions when they will be tangential Poisson actions. EXAMPLE 5.2. Consider the real special linear group
;) a,b,c,d~N, ad-bc=l].
SL(2, I R ) = [ ( ~ We denote el = ~
° o) 1
,
e2 = 2
1
:)
,
e3=~
1(: :)
sl(2, R). Then
is a basis of its Lie algebra [el, e2] = e3,
[e2, e3] = el,
[e3, el] = - e 2 .
Let A = )~1el A e2 -t- )~2e2 A e3 + )~3e3 A el, where )~1, )~2, )~3 c R. Clearly it is an r-matrix of sl(2, R) since A3s/(2, N[) is 1-dimensional and the elements of SL(2, N) have determinants 1. Let a - ( g ) = Lg, A - Rg.A with g ~ SL(2, IR). Then (SL(2, IR), zr) is a Poisson Lie group. The dual Lie bracket is defined by [~, 7/]. = ad~¢ t l - a d ~ ~ for any ~, ~ 6 s/*(2, R). Let e 1,* e 2,* e 3. be the basis of s/*(2, N) dual to el, e2, e3. It is easy to check that the Lie brackets for vectors of this basis are given by [e~, e~] = -)~3e~ - ~.2e~,
[e~, e~] = )~le~ - )~3e~,
[e 3, ell = )~2e~ - )~le~.
Let zrR2 = h(xl, x2)Oxl /x Ox2. Then (R 2, rr•2) be a Poisson manifold for any h(xl, x2) 6 C ~ ( R 2 ) . The natural action of SL(2, N) on IR2 is not transitive and has two orbits: a fixed point orbit O1 = {0} and an open orbit 0 2 = 11{2 - {0}. In order that the natural action ~r : SL(2, 1R) x (R 2, zra2) ~ (IR2, zra2) is a Poisson action, we must have rq~2(gx) = Crg,rC~2(x) + ~rx,Tr(g). (5.7) Let g =
(axa2)
E SL(2, IR),
a3
a4
x =
(xlt x2
ALMOST HOMOGENEOUS POISSON MANIFOLDS
107
Then (5.7) could be rewritten as
h(alxm + a2x2, a3xl +
a4x2) -
h(xi,
x2)
= 1{[(~.1 -~- ~.3)a 2 - - ()L1 - - )~3)a 2 - - 2 ) ~ 2 a l a 3 - - ()~1 + ~.3)] x 2
-'[- [(~.1 -1- ~.3)a2 -- (~q -- ~-3)a2 -- 2)~2a2a4 + (X1 -- ~.3)]x 2 + [2(;.1 + ~.3)ala2 -- 2(L1 -- ~.3)a3a4 --
4L2a2a3]xlx2}.
(5.8)
The solution of (5.8) is h(Xl, x2) =
1
~()~1 "-~ )~3) x 2 -
1
~ (~'1 - )~3)x 2 -
The infinitesimal action of the natural action of 1
(el)p
-m-
1 ~.2XlX2 + C,
SL(2, R)
1
~(XlOxl AX2Ox2),
(e2)p =
C e R.
(5.9)
on R 2 is 1 (e3)p = -~(XlOx 2 AX2Oxl),
~(XlOx2 AX2Oxl),
so this action is a tangential Poisson action if and only if ~.1, )~2, )~3, c satisfy +
> 0,
+
-
< 0,
c > 0.
(5.10)
For example, if ~.1 = ~2 -~- 0 and )~3 = 4, then h(xl, x 2 ) = x 2 + x 2 + c , the natural action a : SL(2, ~ ) × (I~2, rc~2) ~ (R 2, 7r~2) is a tangential Poisson action if c > 0. In the following, we consider the case )~1 = )~3 = 0 and ~'2 = 2 and take h(xl, x 2 ) = c- xlx2. In this case the dual Poisson Lie group SL*(2, ~ ) is realized explicitly by [12]:
SL,(2,~)defSB(2,]~)def{( ~ b+iC-1 ~a ] a > O,b,c ~ ] R , i 2 =
-1}
The dual basis is realized by
• (o
e1 =
,
,(::)
e2 =
and the dual Lie brackets are given by [e 1, e 2]
=
--2e 2,
[e2, e 3] = 0,
[e~, e~] = 2e~.
Note that (el) ° = Spana{e2, e3} is an ideal of sb(2, N), so H = Diag(a, a -1) is a Poisson Lie subgroup of SL(2, R) (cf. [18]). Each orbit of the restricted action of H on R 2 is of the form {(x, y) ~ R21xy = or}, where ot is any constant. Clearly this action preserves the Poisson structure and is tangential, it has a family of momentum maps:
m H : e 2 ---->~'~, ( X l ,
x 2 ) --+
f l 4 l c --
clearly they are H-equivariant momentum maps.
XlX2l ,
Vfl E I[~,
108
Q. YANG
Acknowledgements Part of this work was done while I was visiting as a guest fellow the Institut ftir Mathematik, Ruhr Universit~it Bochum, Germany. I would like to thank Prof. A. Huckleberry and E Heinzner for their financial supports. I also wish to thank Prof. Z.-J. Liu and Prof. J.-H. Lu for many useful comments. Finally, I would like to thank the referee for helpful comments and suggestions for this second version. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]
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