POISSON STRUCTURES ON AFFINE SPACES AND FLAG VARIETIES. I. MATRIX AFFINE POISSON SPACE K. A. Brown, K. R. Goodearl, and M. Yakimov Abstract. The standard Poisson structure on the rectangular matrix variety M (C) is m,n investigated, via the orbits of symplectic leaves under the action of the maximal torus T ⊂ GL (C). Theseorbits, finiteinnumber, areshowntobesmoothirreduciblelocallyclosed m+n subvarieties of M (C), isomorphic to intersections of dual Schubert cells in the full flag m,n variety of GL (C). Three different presentations of the T-orbits of symplectic leaves in m+n M (C) are obtained – (a) as pullbacks of Bruhat cells in GL (C) under a particular m,n m+n map; (b) in terms of rank conditions on rectangular submatrices; and (c) as matrix products of sets similar to double Bruhat cells in GL (C) and GL (C). In presentation (a), the orbits m n of leaves are parametrized by a subset of the Weyl group S , such that inclusions of m+n Zariski closures correspond to the Bruhat order. Presentation (b) allows explicit calculations of orbits. From presentation (c) it follows that, up to Zariski closure, each orbit of leaves is a matrix product of one orbit with a fixed column-echelon form and one with a fixed row- echelon form. Finally, decompositions of generalized double Bruhat cells in M (C) (with m,n respect to pairs of partial permutation matrices) into unions of T-orbits of symplectic leaves are obtained. Introduction 0.1. We investigate the geometry of the affine variety M = M (C) of complex m×n m,n m,n matrices in relation to its standard Poisson structure (see §1.5) and to the action of the torus of “row and column automorphisms”. Specifically, let T denote the torus of diagonal matrices in GL , identified with T ×T where T denotes the corresponding torus in m+n m n ‘ GL . There is a natural action of T on M which arises as the restriction of the natural ‘ m,n left action of GL ×GL on M : namely, (a,b).x = axb−1 for (a,b) ∈ T and x ∈ M . m n m,n m,n This action of T on M is by Poisson isomorphisms; in particular, the action of each m,n element of T maps symplectic leaves of M to symplectic leaves. Thus, it is natural m,n 2000 Mathematics Subject Classification. 53D17; 14L35, 14M12, 14M15, 20G20. The research of the first two authors was partially supported by Leverhulme Research Interchange Grant F/00158/X, and was begun during their participation in the Noncommutative Geometry program at the Mittag-Leffler Institute in Fall 2003. They thank the Institute for its fine hospitality. The research of the second author was also partially supported by National Science Foundation grant DMS-9970159. The research of the third author was partially supported by NSF grant DMS-0406057 and a UCSB junior faculty research incentive grant. He thanks the University of Hong Kong for the warm hospitality during a part of the preparation of this paper. Typeset by AMS-TEX 1 2 K. A. BROWN, K. R. GOODEARL, AND M. YAKIMOV to look at T-orbits of symplectic leaves of M , which are regular Poisson submanifolds m,n of M , rather than at individual symplectic leaves. (Here and throughout, we view the m,n S T-orbit of a symplectic leaf L as the set-theoretic union t.L, rather than as the family t∈T (t.L) of symplectic leaves.) As advantages to this approach, we mention that T-orbits t∈T of symplectic leaves are easier to identify than single symplectic leaves, and these orbits exhibit direct relations with known geometric and Lie-theoretic structures. For example, we prove that the T-orbits of symplectic leaves in M are isomorphic (as varieties) to m,n intersections of dual Schubert cells in the full flag variety of GL , and each generalized m+n double Bruhat cell in M (corresponding to a pair of partial permutation matrices) is a m,n disjoint union of T-orbits of symplectic leaves, containing one such orbit as an open dense subset. One thus sees that the Poisson structure of M is in some ways similar to, but m,n also more intricate than, that of the group GL – for instance, as follows from the analysis n of Hodges and Levasseur [15], the orbits of symplectic leaves in GL under left translation n by the standard maximal torus are precisely the double Bruhat cells. In a sequel to this paper, we will investigate the relation between the standard Poisson structures on different (partial) flag varieties of a semisimple algebraic group. We will also relate the restriction of the Poisson structure to various Poisson subvarieties with known quadratic Poisson structures on affine spaces. A detailed study of Poisson structures of the latter type associated to arbitrary Schubert cells in flag varieties of semisimple groups will be presented as well. 0.2. Recall that a Poisson group structure on an algebraic group G is thought of as the “semiclassical limit” of a quantization of G, a viewpoint promulgated in particular by Drinfeld and his school (cf. [6]). Relationships between the symplectic foliation of such a Poisson structure and the primitive spectrum of a quantized coordinate ring O (G) are q viewed under the heading of a generalized version of the Kirillov–Kostant orbit method. In the work of Soibelman (e.g., [27]) on compact groups G, this led to bijections between the symplectic leaves of G and the primitive ideals of O (G). Hodges and Levasseur [15, q 16] then established analogous results for G = SL , which were extended to all semisimple n groups by Joseph [19]. We take the corresponding viewpoint that the Poisson structure on M is the semiclassical limit of the structure of O (M ), and argue that the results of m,n q m,n the present paper should correspond to the framework of the primitive ideals in O (M ). q m,n Specifically, we conjecture that the sets of minors which define the T-orbits of symplectic leaves in M (obtainable from Theorem 4.2) should match the sets of quantum minors m,n which generate the prime ideals of O (M ) invariant under winding automorphisms (cf. q m,n [13]). Some relations between M and O (M ) are already known. In particular, m,n q m,n the set of T-orbits of symplectic leaves in M , partially ordered by inclusions of clo- m,n sures, is anti-isomorphic to the poset T-SpecO (M ) of winding-invariant prime ideals q m,n in O (M ) – our work shows that the former poset is anti-isomorphic to the set q m,n n (cid:16) (cid:17)o ≤wm,n (cid:12) 1 2 ··· n n+1 n+2 ··· n+m Sm+◦n = y ∈ Sm+n (cid:12) y ≤ m+1 m+2 ··· m+n 1 2 ··· m ≤wm,n under the Bruhat order, while Launois [22, Theorem 5.6] has proved that S ◦ is iso- m+n morphic to T-SpecO (M ). q m,n THE MATRIX AFFINE POISSON SPACE 3 0.3. Before summarizing our main results, we indicate some notation, beginning with N = m + n. By Gr(n,N) we denote the Grassmannian of n-dimensional subspaces of an N-dimensional space. We write B+ and B− for the standard Borel subgroups of any ‘ ‘ GL (consisting of upper, respectively lower, triangular matrices), and identify the Weyl ‘ group of GL with both the symmetric group S and the group of permutation matrices ‘ ‘ in GL . The symbol w‘ denotes the longest element of S . For 0 ≤ t ≤ ‘, let S1 and ‘ ◦ ‘ t S2 denote the natural copies of S and S inside S , acting on the numbers 1,...,t and ‘−t t ‘−t ‘ t + 1,...,‘, respectively. Finally, for a Weyl group W and a subgroup W generated by 1 simple reflections, WW1 denotes the set of minimal length representatives for left cosets in W/W . RecallthateachsuchcosethasauniquerepresentativeinWW1 (cf. [3, Proposition 1 2.3.3(i)]). The following theorem summarizes Theorems 3.9 and 3.13. Theorem A. (a) There are only finitely many T-orbits of symplectic leaves in M , and m,n they are smooth irreducible locally closed subvarieties. (b) The T-orbits of symplectic leaves in M can be described as the sets m,n (cid:26) (cid:12) (cid:27) (cid:12) hwn 0 i Pw = x ∈ Mm,n (cid:12)(cid:12) x◦ w◦m ∈ BN+wBN+ , h i wn 0 where w ∈ S and w ≥ ◦ in the Bruhat order. N 0 wm ◦ (c) The closure of P equals the disjoint union of the P for z ≤ w. w z (d) As an algebraic variety, P is isomorphic to the intersection of dual Schubert cells w hwn 0 i B−. ◦ B+ ∩B+.wB+ N 0 wm N N N ◦ in the full flag variety GL /B+. (cid:3) N N 0.4. Fulton [10] has given computational descriptions of Bruhat cells B+wB+ in terms N N of ranks of rectangular submatrices. We apply his results to the sets P , to characterize w exactly which matrices x lie in each P , in terms of ranks of rectangular submatrices of x. w See Theorem 4.2 for the precise statement. 0.5. The results of Theorem A are obtained by embedding M in the Grassmannian m,n Gr(n,N) which, equipped with an appropriate Poisson structure, becomes a Poisson ho- mogeneous space for the standard Poisson algebraic group GL . (For details on Poisson N homogeneous spaces for Poisson algebraic groups see [7] or Section 1.) This approach provides, in addition, a natural Poisson compactification of M which, in particular, m,n suggests an approach to the problem of studying the spectrum of O (M ) via noncom- q m,n mutative projective geometry. A completely different viewpoint is obtained by focussing, as we do in Sections 5 and 6, on the sets Om,n of matrices in M with a fixed rank t. Each Om,n is a Poisson homoge- t m,n t neous space for the natural action of GL ×GL (equipped with an appropriate Poisson m n group structure). The latter group is the Levi factor of the maximal parabolic subgroup of GL defining Gr(n,N). The key results of this approach, taken from Theorems 5.11, N 6.1, 6.4 and Corollary 6.5, are as follows. 4 K. A. BROWN, K. R. GOODEARL, AND M. YAKIMOV Theorem B. Fix a nonnegative integer t ≤ min{m,n}. (a) The T-orbits of symplectic leaves within Om,n can be described as the sets t Pt = [ (cid:0)B+yB+ ∩B−zτ(cid:1).hIt 0i.(cid:0)τ−1B−u−1B− ∩v−1B+(cid:1), (y,v,z,u) m m m 0 0 n n n τ∈S1 t zτ≤y, vτ−1≤u S2 S1 S1 S2 where (y,v,z,u) ∈ S m−t ×S t ×S t ×S n−t and z ≤ y, v ≤ u. m n m n (b) For (y,v,z,u) as in (a), the set h i C .R = (cid:0)B+yB+ ∩B−z(cid:1). It 0 .(cid:0)B−u−1B− ∩v−1B+(cid:1) y,z u,v m m m 0 0 n n n is dense in Pt . (y,v,z,u) h i (c) The sets C = (cid:0)B+yB+ ∩ B−z(cid:1). It are (T × T )-orbits of symplectic leaves y,z m m m 0 m t of M , and each of the sets consisting of all matrices in M with rank t and a given m,t m,t column-echelon form is a disjoint union of certain C . y,z (cid:0) (cid:1) (d) The sets R = [I 0]. B−u−1B− ∩ v−1B+ are (T × T )-orbits of symplectic u,v t n n n t n leaves of M , and each of the sets consisting of all matrices in M with rank t and a t,n t,n given row-echelon form is a disjoint union of certain R . (cid:3) u,v The descriptions of torus orbits of symplectic leaves in M given in part (b) of The- m,n orem A and part (a) of Theorem B are matched in Theorem 5.11 and Proposition 5.9. 0.6. Finally, we study the decomposition of M into generalized double Bruhat cells m,n Bw1,w2 = B+w B+ ∩B−w B−, m 1 n m 2 n for partial permutation matrices w , w . If w and w have the same rank t (which is 1 2 1 2 necessary for Bw1,w2 to be nonempty), there are unique decompositions h i h i w = y It 0 v−1 w = z It 0 u−1 1 0 0 2 0 0 S2 S1S2 S1S2 S2 where y ∈ S m−t, v ∈ S t n−t, z ∈ S t m−t, and u ∈ S n−t (see Lemma 7.3). The m n m n following results are given in Theorem 7.4. Theorem C. Let w ,w ∈ M be partial permutation matrices with rank t, decomposed 1 2 m,n as above. (a) Bw1,w2 is nonempty if and only if z ≤ y and v ≤ u, in which case it is a T-stable Poisson subvariety of M , and a smooth irreducible locally closed subvariety. m,n (b) The partition of Bw1,w2 into T-orbits of symplectic leaves is given by (cid:26) (cid:12) (cid:27) Bw1,w2 = G Pt (cid:12)(cid:12) τ1 ∈ Sm2 −t ⊆ Sm, zτ1 ≤ y . (y,vτ2,zτ1,u) (cid:12) τ2 ∈ Sn2−t ⊆ Sn, vτ2 ≤ u (c) Pt is Zariski open and dense in Bw1,w2. (cid:3) (y,v,z,u) THE MATRIX AFFINE POISSON SPACE 5 0.7. Let us also note that the standard Poisson algebraic group GL is a T-stable Poisson m subvariety of M . Thus the T-orbits of symplectic leaves of GL (which are the same m,m m as the T -orbits of leaves) comprise a subset of the T-orbits of symplectic leaves of M . m m,m The former are the double Bruhat cells B+w B+ ∩B−w B− of GL , for w ,w ∈ S . m 1 m m 2 m m 1 2 m They were studied in detail by Fomin and Zelevinsky in [8], who in a joint work with Berenstein also proved [1] that their rings of regular functions provide important examples of upper cluster algebras [9]. Our results in particular show that the double Bruhat cells in GL are special cases of intersections of dual Schubert cells on the full flag variety m of GL . It would be very interesting to understand whether any intersection of dual 2m Schubert cells on the full flag variety of an arbitrary reductive algebraic group gives rise to a cluster algebra in the sense of Fomin and Zelevinsky [9]. If this is true, it will imply that any T-orbit of symplectic leaves of M is the spectrum of a cluster algebra. m,n 0.8. We conclude the introduction with some remarks on our notation and conventions. All manifolds and algebraic varieties considered in this paper are over the field of complex numbers. Given an algebraic group G with tangent Lie algebra g, we denote by L(γ) and R(γ) the left and right invariant multi-vector fields on G corresponding to γ ∈ ∧g. If G acts on a smooth quasiprojective variety M, we will denote by (0.1) χ : ∧g → Γ(M,∧TM) the extension of the infinitesimal action of g on M to ∧g. In the special case of the left and right multiplication actions of G on itself (g.a = ga and g.a = ag−1), the above infinitesimal actions will be denoted by (0.2) χR,χL : ∧g → Γ(G,∧TG). Note that for γ ∈ ∧g, (0.3) χL(γ) = R(γ) χR(γ) = (−1)(cid:15)(γ)L(γ), where (cid:15)(γ) is the parity of γ. If Y is a locally closed subvariety of an algebraic variety X and Z ⊆ Y, we will denote the closure of Z in Y by Cl (Z). By a stratification of an algebraic variety X we mean Y F a partition of X into smooth, irreducible, locally closed subvarieties, X = X , such α∈A α F that for each α ∈ A, we have X = X for some index set A(α) ⊆ A. α β∈A(α) β We will use the following convention to distinguish double cosets from orbits of cosets. For any subgroups C and D of a group G: (i) The (C,D) double coset of g ∈ G will be denoted by CgD; (ii) The C-orbit of gD in G/D will be denoted by C.gD. The adjoint action of g ∈ G on h ∈ G will be written as Ad (h) = ghg−1. g 6 K. A. BROWN, K. R. GOODEARL, AND M. YAKIMOV 1. Poisson algebraic groups and Poisson homogeneous spaces We begin with background and notation for Poisson algebraic groups and Poisson ho- mogeneous spaces, and then characterize the symplectic leaves and their orbits in certain Poisson homogeneous spaces. 1.1. Poisson varieties. Recall that a Poisson manifold is a pair (X,π) consisting of a smooth manifold X together with a Poisson bivector field π ∈ ∧2TX, that is, [π,π] = 0 where [.,.] denotes the Schouten bracket. A (not necessarily closed) submanifold Y of X is called a Poisson submanifold if π ∈ ∧2T Y for all y ∈ Y. In this case (Y,π| ) is a Poisson y y Y manifold as well. A (not necessarily closed) submanifold Y of X is called a complete Poisson submanifold if it is stable under all Hamiltonian flows. Any complete Poisson submanifold is a Poisson submanifold. The converse is not necessarily true but, if (X,π) is a Poisson manifold which is partitioned into a disjoint union of Poisson submanifolds F X = Y , then all Y are complete Poisson submanifolds, see [17, Lemma 3.2]. α∈A α α The Poisson manifold (X,π) is regular, respectively symplectic, if rankπ is constant, respectively rankπ = dimX. A symplectic leaf of (X,π) is a maximal connected (not necessarily closed) symplectic submanifold. It is well known that any Poisson manifold (X,π) can be decomposed into a disjoint union of its symplectic leaves, see e.g. [29, 28]. Note that a (not necessarily closed) submanifold Y of X is a complete Poisson submanifold if and only if it is a union of symplectic leaves of (X,π). Let us also recall that a map φ : (X,π) → (Z,π0) between two Poisson manifolds is called a Poisson map if φ (π ) = π0 for all x ∈ X. For instance if Y is a Poisson ∗ x φ(x) submanifold of (X,π), the natural inclusion i : (Y,π| ) ,→ (X,π) is Poisson. Y All Poisson manifolds considered in this paper will be (complex) smooth quasiprojective Poisson varieties. The symplectic leaves of a smooth quasiprojective Poisson variety are not necessarily algebraic, i.e., smooth irreducible locally closed subvarieties. We will see below that this is the case for many Poisson varieties admitting appropriate transitive algebraic group actions. 1.2. Poisson algebraic groups and Manin triples. A Poisson algebraic group is an algebraic group G equipped with a Poisson bivectorfield π ∈ ∧2TG such that the map (G,π)×(G,π) → (G,π) is Poisson. The tangent Lie algebra g = Lie(G) of a Poisson algebraic group (G,π) has a canonical Lie bialgebra structure; see [4, §1.3] and [21, §3.3] for details. RecallthataManin triple of Lie algebras isatriple(d,a,b)withthefollowingproperties: (1) d is a Lie algebra, a and b are Lie subalgebras of d, and d is the vector space direct sum of a and b. (2) d is equipped with a nondegenerate invariant bilinear form with respect to which both a and b are Lagrangian (i.e., maximal isotropic) subspaces. To any Lie bialgebra g one associates the Manin triple (D(g),g,g∗). Here D(g) and g∗ are the underlying Lie algebras of the double and the dual Lie bialgebras of g. The bilinear form on D(g) is given by hx+α,y +βi = β(x)+α(y) for x,y ∈ g, α,β ∈ g∗. THE MATRIX AFFINE POISSON SPACE 7 1.3. Definition. A Manin triple of algebraic groups is a triple (D,A,B) of algebraic groups such that A and B are algebraic subgroups of D and (Lie(D),Lie(A),Lie(B)) is a Manin triple of Lie algebras. Fix a Manin triple of algebraic groups (D,A,B). Then D has a canonical Poisson algebraic group structure with a Poisson bivector field given by X πD = L(r)−R(r) = χR(r)−χL(r) where r = x ∧xi ∈ ∧2LieD i i in the notation (0.2)–(0.3) for left and right invariant multi-vector fields L(.), R(.) on D and infinitesimal actions χL(.), χR(.) of LieD on D. Here {x } and {xi} are dual bases i of Lie(A) and Lie(B), respectively, with respect to the nondegenerate bilinear form on Lie(D). The groups A and B are Poisson subvarieties of D. The Poisson algebraic group (D,πD) is a double of (A,πD| ), and (B,−πD| ) is a dual Poisson algebraic group of (A,πD| ); A B A cf. [4, §1.4] and [21, §3.3]. We will say that a Poisson algebraic group (G,π) is a part of a Manin triple of algebraic groups (D,G,F) if the Poisson structure π coincides with the Poisson structure πD| G induced from D. 1.4. Standard Poisson structures on reductive algebraic groups. Let G be a com- plex reductive algebraic group. The standard Poisson structure on G, turning it into a Poisson algebraic group, is defined as follows. Fix two opposite Borel subalgebras b± of g = LieG and set h = b+ ∩b− for the corresponding Cartan subalgebra of g. Fix a non- degenerate bilinear invariant form h.,.i on g for which the square of the length of a long root is equal to 2. Choose sets of root vectors {e } and {f }, spanning respectively the α α nilradicals n+ and n− of b+ and b−, normalized by he ,f i = 1. α α The standard r-matrix of g is given by X (1.1) r = e ∧f α α α and the corresponding standard Poisson structure on G is defined by (1.2) π = L(r)−R(r) = χR(r)−χL(r), in the notation (0.2)–(0.3). The standard r-matrix on G = GL is N X (1.3) rN = E ∧E ∈ ∧2gl ij ji N 1≤i<j≤n where the E are the standard elementary matrices. ij By abuse of notation, GL will denote the algebraic group GL equipped with the N N standard Poisson structure πN from (1.2), associated to the r-matrix rN (1.3). By GL• N we will denote the Poisson algebraic group (GL ,−πN). N 8 K. A. BROWN, K. R. GOODEARL, AND M. YAKIMOV Any standard (complex) reductive Poisson algebraic group (G,π) is a part of the Manin triple (G×G,∆(G),F) where ∆(G) is the diagonal of G×G and (1.4) F = {(hu+,h−1u−) | h ∈ T, u± ∈ U±} ⊆ B+ ×B−, where B± are the Borel subgroups of G corresponding to b±, U± are their unipotent radicals, and T = B+ ∩ B− is the corresponding maximal torus of G. For the standard Poisson structure on G, (1.5) g∗ = LieF = {(h+n+,−h+n−) | h ∈ h, n± ∈ n±} ⊆ b+ ⊕b−. ∼ The nondegenerate invariant bilinear form on Lie(G×G) = g⊕g, used in the Manin triple of Lie algebras (g⊕g,∆(g),g∗), is (1.6) h(x ,x ),(y ,y )i = hx ,y i−hx ,y i, 1 2 1 2 1 1 2 2 where in the right hand side h.,.i denotes the bilinear form on g, fixed above. 1.5. Matrix affine Poisson spaces. The m×n matrix affine Poisson space is the affine space Amn, identified with the space M of all m×n complex matrices. The standard m,n Poisson structure on M is given by m,n m n (1.7) πm,n = X X(cid:0)sign(k −i)+sign(l−j)(cid:1)x x ∂ ∧ ∂ il kj ∂x ∂x ij kl i,k=1j,l=1 intermsofthestandardcoordinatefunctionsx onM . By abuse of notation, M will ij m,n m,n denote the matrix affine Poisson space, thus dropping the symbol for the Poisson structure (1.7) on M . m,n Note that GL acts on M by left multiplication (g.x = gx for g ∈ GL , x ∈ M ), m m,n m m,n and GL acts on M by (inverted) right multiplication (g.x = xg−1 for g ∈ GL , n m,n m x ∈ M ). The extensions of the corresponding infinitesimal actions of gl and gl on m,n m n M to ∧gl and ∧gl will be denoted by n m n χL : ∧gl → Γ(M ,TM ) and χR : ∧gl → Γ(M ,TM ). m m,n m,n n m,n m,n Note that in the case m = n these extend the infinitesimal actions χL and χR of gl on m GL ⊆ M , defined in (0.2). m m By direct computation one shows that the Poisson structure (1.7) on M is also given m,n by the formula (1.8) πm,n = χR(rn)−χL(rm) in terms of the standard r-matrix rN for GL , see (1.3). N Note that GL is a Poisson subvariety of M . n n,n THE MATRIX AFFINE POISSON SPACE 9 1.6. Poisson homogeneous spaces. Fix a Poisson algebraic group (G,π) and set g = Lie(G). A Poisson (G,π)-space is a smooth quasiprojective Poisson variety (M,π ) M equipped with a morphic G-action for which (G,π)×(M,π ) → (M,π ) M M is a Poisson morphism. A Poisson homogeneous space for (G,π) is a Poisson (G,π)-space (M,πM) for which M is a homogeneous G-space. (Recall that any homogeneous space of an algebraic group is a smooth quasiprojective variety [2, Theorem 6.8].) To each m ∈ M, one associates the Drinfeld subalgebra [7] l = {x+α ∈ D(g) | x ∈ g, α ∈ g∗, α| = 0, αcπM = x+g } m gm m m of the double D(g), where g denotes the Lie algebra of the stabilizer G = Stab (m), the m m G tangent space T M is identified with g/g , and the Poisson bivectorfield πM is thought m m m of as an element of ∧2(g/g ). Note that m (1.9) g = g∩l . m m The Drinfeld subalgebras l are moreover Lagrangian subalgebras of the double D(g), m equipped with the canonical nondegenerate invariant bilinear form [7; 21, Proposition 6.2.15]. The map, associating to m ∈ M its Drinfeld subalgebra l ⊆ D(g), is G- m equivariant: l = Ad (l ) gm g m where Ad refers to the adjoint action of G on D(g). g 1.7. Definition. A Poisson homogeneous (G,π)-space (M,πM) will be called algebraic if the Drinfeld subalgebra of some m ∈ M is the tangent Lie algebra of an algebraic subgroup L ⊆ D. m Because of the G-equivariance of the map m 7→ l , if the condition in the definition is m satisfied for one point m ∈ M, then it holds for any m ∈ M. Animportanttype ofPoissonhomogeneous space (M,πM)is the classofthose forwhich πM vanishes at some point of M. In the rest of this subsection we describe those. An algebraic subgroup Q of a Poisson algebraic group (G,π) will be called an almost Poisson algebraic subgroup if π ∈ T Q∧T G q q q for all q ∈ Q. (Recall that if π ∈ ∧2T Q for all q ∈ Q, then Q is called a Poisson algebraic q q subgroup of (G,π).) Fix an almost Poisson algebraic subgroup Q of (G,π), and consider the projection p : G → G/Q, p(g) = gQ. Then π −R (π ) ∈ L (T Q)∧T G gq q g g q gq 10 K. A. BROWN, K. R. GOODEARL, AND M. YAKIMOV for all g ∈ G, q ∈ Q, and the rule G/Q (1.10) π = p (π ), g ∈ G gQ ∗ g gives a well-defined Poisson structure πG/Q on G/Q. The pair (G/Q,πG/Q) is a Poisson homogeneous space of (G,π) and πG/Q vanishes at the base point eQ of G/Q. 1.8. Theorem. Fix a Poisson algebraic group (G,π). (a) Any Poisson homogeneous (G,π)-space (M,πM) with the property that the Poisson bivectorfield πM vanishes at some point m ∈ M is isomorphic to (G/Q,πG/Q) for Q = Stab (m) which is an almost Poisson algebraic subgroup of (G,π). G (b) For an almost Poisson algebraic subgroup Q of G, the Drinfeld Lagrangian subalgebra of the base point eQ of the Poisson homogeneous space (G/Q,πG/Q) is (1.11) l = q+q⊥ where q = LieQ and q⊥ refers to the orthogonal subspace to q ⊆ g in g∗. (c) A connected algebraic subgroup Q of (G,π) is an almost Poisson algebraic subgroup if and only if the orthogonal complement q⊥ ⊆ g∗ is a subalgebra of the dual Lie bialgebra g∗ of g (as in part (b), q = LieQ). (d) A connected algebraic subgroup Q of (G,π) is a Poisson algebraic subgroup if and only if q⊥ is an ideal in g∗. (cid:3) Parts (a) and (d) of this theorem can be found, e.g., in [21, page 52 and Proposition 6.2.3]; parts (b) and (c) are well known. Below we gather some results on symplectic leaves of algebraic Poisson homogeneous spaces. Fix a Poisson algebraic group (G,π) which is a part of a Manin triple of algebraic groups (D,G,F), as defined in §1.3. Fix also an algebraic Poisson homogeneous (G,π)- space with connected stabilizer subgroups G (see §1.6). Such a homogeneous space m has the form G/N where N is a connected subgroup of G and the Drinfeld Lagrangian subalgebra of Lie(D) corresponding to the base point eN ∈ G/N integrates to an algebraic subgroup L ⊆ D. Note that N = (G∩L)◦, the identity component of G∩L, because of (1.9) and the connectedness of N. Consider the composition of maps µ ∼= (1.12) Π : G/N −→ G/(G∩L) −→ G.L ⊆ D/L, where µ is the map gN 7→ g(G∩L). 1.9. Theorem. Assume that (G,π) is a Poisson algebraic group which is a part of a Manin triple of algebraic groups (D,G,F). Let (G/N,π0) be an algebraic Poisson homo- geneous (G,π)-space with connected stabilizer subgroups for which the Drinfeld Lagrangian subalgebra of the base point eN is LieL for an algebraic subgroup L of G.