Covariant Poisson structures on complex Grassmannians

arXiv:math/0503218v1 [math.SG] 11 Mar 2005

Covariant Poisson structures on complex Grassmannians N. Ciccoli ∗ Dipartimento di Matematica e Informatica Universitá di Perugia A. J.–L. Sheu † Department of Mathematics University of Kansas February 1, 2008

Abstract The purpose of this paper is to study covariant Poisson structures on Grkn C obtained as quotients by coisotropic subgroups of the standard Poisson–Lie SU (n). Properties of Poisson quotients allow to describe Poisson embeddings generalizing those obtained in [Sh3].

1

Introduction

In [Sh3] a family of covariant Poisson structures on complex projective spaces underlying the Dijkhuizen–Noumi quantization ([DiNo]) was studied from the point of view of coisotropic subgroups with respect to an affine Poisson structure on SU (n), providing also a description of the associated Lagrangian subalgebras and Poisson embeddings of standard odd Poisson spheres in non standard Poisson projective spaces. In this paper we plan to extend those results to complex Grassmannians (their quantum version may be found in [NDS]). The emphasis is laid even more strongly on the role played by subgroups which are coisotropic with respect to the standard multiplicative Poisson structure on SU (n). ∗

supported by PRIN Azioni di gruppi su variet´ a and GNSAGA. supported by the University of Kansas General research Fund allocation #2301 for FY 2004. †

1

One reason of interest lies in the fact that every coisotropic subgroup of a Poisson–Lie group can be quantized in such a way as to fit in a nice duality diagram ([CG]). Furthermore coisotropic submanifolds have recently raised a lot of attention in the context of deformation quantization ([CF, BGHW]) and played a role in the analysis of Poisson sigma–models over group manifolds ([BZ]). In the first section we clarify the relation between coisotropic subgroups of a Poisson–Lie group and coisotropic subgroups of translated affine Poisson bivectors. This relates results in [Sh3] with those in the present work, allowing a natural interpretation from the foliation point of view. In the second section we describe the family of covariant Poisson structures on complex Grassmannians under consideration and show how it can be obtained as quotients by coisotropic subgroups. Such structure was first introduced in [KhRaRu] under different methods. A specific non standard Grassmannian was studied, recently, with Lie group methods by Foth and Lu (see [Flu]). Finally in the last section we describe a general procedure allowing to determine Poisson embeddings of G–spaces. Applying it to projective spaces we show how it recovers the whole symplectic foliation in the standard case and the Poisson embeddings of [Sh3] in the nonstandard one. Moving on to Grassmannians such procedure will give embeddings of standard Poisson– Stiefel manifolds (and of other more general manifolds) in non standard 2m (C) this Poisson Grassmannians. In the special case of Grassmannians Grm will result in a Poisson embedding of the standard Poisson–Lie group U (m). Such embeddings are relevant also from the point of view of quantum spaces, where they were first identified. It is in the study of the groupoid C ∗ – algebra C(Pnq,c ) carried out in [Sh1, Sh2], in fact, they were used to construct composition sequences for the algebra and, eventually, to compute its K– groups. We expect that quantum Stiefel manifolds studied in [PV] and suitable generalizations will appear as quotients of nonstandard complex q–Grassmannians and allow a similar detailed analysis.

2 2.1

Coisotropic and affine coisotropic subgroups Affine Poisson structures

Let G be a given Lie group, with Lie algebra g. In the following, we use Rg (resp. Lg ) to denote the right (resp. left) translation action on G by g ∈ G, and also all the actions induced by it on tensors of G, e.g. Rg (v) = (D (Rg ))x (v) ∈ Txg G for any vector v ∈ Tx G, and (Rg X) (x) = 2

(D (Rg ))xg−1 X xg−1 ∈ Tx G for any vector field X ∈ Γ (T G), where D (Rg ) is the differential (a vector bundle map on T G) of the diffeomorphism Rg . Note that the right translation R is an anti-homomorphism, i.e. Rg Rh = Rhg . Similarly we have the left translation Lx , but L is a homomorphism, i.e. Lg Lh = Lgh . Given any 2–tensor ρ : G → ∧2 T G let 2 ρ˜(g) := L−1 g ρ(g) ∈ ∧ g. First we recall the following facts for an (alternating) 2-tensor field ρ on a Lie group G (see [Lu, We1]). (1) ρ is called multiplicative if ρ (gh) = Lg (ρ (h)) + Rh (ρ (g)) . (Note that ρ (e) = 0 if ρ is multiplicative, where e is the unit of G.) (2) ρ is called affine if ρ (gh) = Lg (ρ (h)) + Rh (ρ (g)) − Lg Rh (ρ (e)) . (3) ρ is affine if and only if π := ρ − (ρ (e))l is multiplicative, where X l denotes the left-invariant tensor field generated by X ∈ ∧2 g. (Note that for any 2-tensor field π with π(e) = 0 and X ∈ ∧2 g, if ρ := π +X l , then ρ (e) = X and hence π = ρ − (ρ (e))l . So all affine ρ are of the form ρ = π + X l for some multiplicative π and X ∈ ∧2 g.) (4) If π is a Poisson-Lie structure on G, then ρ = π + X l (with ρ (e) = X) is affine for any X ∈ ∧2 g (but may not be Poisson); in this case ρ is also Poisson if and only if dX = 12 [X, X] ([DaSo]). (5) If ρ is affine Poisson, then π := ρ − (ρ (e))l is multiplicative and Poisson. (But the converse may not be true, cf. (4) above.) (6) Given ρ Poisson, we have that ρ is affine Poisson if and only if π := ρ − (ρ (e))l is multiplicative Poisson (or Poisson-Lie). Now we show that if ρ is affine Poisson and ρ (σ) = 0 for some point σ ∈ G, then Rσ−1 ρ is Poisson-Lie. Lemma 1 . If ρ is affine Poisson, then Rσ ρ is also affine Poisson for any σ ∈ G. (We don’t assume ρ (σ) = 0 in this lemma.)

3

Proof. Clearly the right translation of a Poisson structure on G is still a Poisson structure. So Rσ ρ is Poisson. Now by the commutativity Rg Lh = Lh Rg for all g, h ∈ G, we get i h Rσ ρ − ((Rσ ρ) (e))l (g) = Rσ ρ gσ −1 − Lg ((Rσ ρ) (e)) = Rσ Lg ρ σ −1 + Rσ−1 (ρ (g)) − Lg Rσ−1 (ρ (e)) − Lg Rσ ρ σ −1 = Rσ Lg ρ σ −1 + ρ (g) − Lg (ρ (e)) − Lg Rσ ρ σ −1 i h l = ρ (g) − Lg (ρ (e)) = ρ − (ρ (e)) (g)

which shows that Rσ ρ − ((Rσ ρ) (e))l = ρ − (ρ (e))l a multiplicative Poisson structure since ρ is affine Poisson. Thus Rσ ρ is affine Poisson. Proposition 2 . If ρ is affine Poisson and ρ (σ) = 0 for some point σ ∈ G, then Rσ−1 ρ is Poisson-Lie. Proof. Rσ−1 ρ is affine Poisson with (Rσ−1 ρ) (e) = Rσ−1 (ρ (σ)) = 0 and hence Rσ−1 ρ is multiplicative Poisson.

2.2

Coisotropic subgroups

In this section we will clarify the relation between affine Poisson structures on Lie groups and coisotropic subgroups of Poisson–Lie groups, introducing the notion of affinely coisotropic subgroup. Recall that for a given Poisson manifold (M, πM ) a coisotropic submanifold is an embedded submanifold such that its defining ideal (i.e. the ideal of smooth functions which are zero on the manifold) is a Poisson subalgebra. For a given Poisson–Lie group (G, π) a coisotropic subgroup is a Lie subgroup H which is also a coisotropic submanifold. At the infinitesimal level, if δ = (D˜ π )e : g → ∧2 g represents the cobracket and h is a Lie subalgebra of g then h can be integrated to a coisotropic subgroup if and only if δ(h) ⊆ h ∧ g. Let ρ be an affine Poisson structure on the Lie group G and let H be a closed (connected) subgroup. It is known that the multiplicative Poisson 4

structure on G induces (or projects to) a well-defined Poisson structure on G/H when H is a coisotropic subgroup. The concept of a coisotropic subgroup H of an affine Poisson Lie group (G, ρ) is more delicate, and there is a fine distinction between “a coisotropic subgroup” and “a subgroup that is a coisotropic submanifold” as discussed below. First we note that the following conditions are equivalent: (2) ρ (h) − Rh (ρ (e)) ∈ Lh (h ∧ g) for all h ∈ H; (3) ρ (kh) − Rh (ρ (k)) ∈ Lkh (h ∧ g) for all h, k ∈ H; (4) ((Dρ˜)e + [˜ ρ (e) , ·]) (h) ⊂ h ∧ g; (5) ρ (gh) − Rh (ρ (g)) ∈ Lgh (h ∧ g) for all h ∈ H and g ∈ G. Furthermore if adh(ρ(e)) ⊆ h ∧ g such conditions are equivalent to (1) H is a ρ-coisotropic submanifold of G, i.e. ρ (h) ∈ Lh (h ∧ g) for all h ∈ H; In fact,

−1 Rh (ρ (e)) = Lh L−1 h Rh (ρ (e)) = Lh Adh (ρ (e))

and so (1) ⇔ (2) if adh (ρ (e)) ⊂ h ∧ g. Since ρ is affine, we have Lg [ρ (h) − Rh (ρ (e))] = ρ (gh) − Rh (ρ (g)) for any h ∈ H and g ∈ G, and hence (2), (3), and (5) are clearly equivalent. From −1 L−1 h [ρ (h) − Rh (ρ (e))] = Lh−1 ρ (h) − Adh (ρ (e)) and D Ad−1 h (ρ (e)) h=e = − ad· (ρ (e)), it is not hard to see the equivalence of (3) and (4). Note that even if ρ is multiplicative and a subgroup H is a ρ–coisotropic submanifold, a coset gH of H in general need not be a ρ-coisotropic submanifold of G, but is “affinely (or relatively) ρ-coisotropic” in the sense of condition (5). Note that in general, when ρ (e) 6= 0, i.e. ρ is not multiplicative, both (1)⇒(2) and (2)⇒(1) may not hold. We define a closed subgroup H of an (affine) Poisson Lie group G to be a ρ-coisotropic subgroup if each coset gH with g ∈ G is an affinely ρ-coisotropic submanifold of G, i.e. ρ (gh) − Rh (ρ (g)) ∈ Lgh (h ∧ g) for all h ∈ H. So when ρ is multiplicative, a closed subgroup H of G is a ρ-coisotropic submanifold of G if and only if H is a ρ-coisotropic subgroup of G.

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Theorem 3 . Let π be a Poisson 2-tensor on a Lie group G. For a closed Lie subgroup H of G and σ ∈ G, the conjugate Adσ H of H is π-coisotropic if and only if H is πσ -coisotropic, where πσ (g) := Rσ π gσ −1 for g ∈ G. 2 Proof. Let π ˜ (g) := L−1 g (π (g)) ∈ ∧ g. Since

−1 −1 L−1 ˜ (gh) − L−1 ˜ (gh) − L−1 gh [π (gh) − Rh π (g)] = π h Lg Rh π (g) = π h Rh Lg π (g)

=π ˜ (gh) − L−1 ˜ (g) = π ˜ (gh) − Ad−1 π (g)) , h Rh π h (˜ a subgroup H is π-coisotropic if and only if π ˜ (gh) − Ad−1 π (g)) ∈ h ∧ g h (˜ for all g ∈ G and h ∈ H. Thus Adσ H is π-coisotropic if and only if π (g)) ∈ Adσ (h) ∧ g (*) π ˜ (g Adσ (h)) − Ad−1 Adσ (h) (˜ for all g ∈ G and h ∈ H. Similarly, H is πσ -coisotropic if and only if (**) π fσ (gh) − Ad−1 πσ (g)) ∈ h ∧ g h (f

for all g ∈ G and h ∈ H. Note that

−1 π fσ (g) = L−1 Rσ π gσ −1 g (πσ (g)) = Lg

−1 = Rσ L−1 g π gσ

−1 −1 −1 −1 = Rσ L−1 ˜ gσ −1 = Rσ L−1 = Rσ L−1 σ π σ Lσ Lg π gσ σ Lgσ−1 π gσ = Ad−1 π ˜ gσ −1 σ

for any g ∈ G. So

Ad−1 π fσ (gh) − Ad−1 πσ (g)) = Ad−1 π ˜ gσ −1 π ˜ ghσ −1 − Ad−1 σ σ h (f h −1 −1 = Ad−1 π ˜ gσ −1 Adσ (h) − Ad−1 π ˜ gσ −1 σ σ Adσ Adh Adσ = Ad−1 π ˜ gσ −1 π ˜ gσ −1 Adσ (h) − Ad−1 σ σhσ−1

and hence the condition (**) is equivalent to ˜ gσ −1 ∈ Adσ (h ∧ g) = Adσ (h) ∧ g π ˜ gσ −1 Adσ (h) − Ad−1 Adσ h π

for all g ∈ G and h ∈ H, or equivalently, the condition (*).

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Proposition 4 Let G be a Poisson–Lie group, H a closed subgroup such that its conjugate Adσ H := Adσ H = σHσ −1 is coisotropic where σ ∈ G. Let πσ be the affine Poisson structure on G given by πσ (g) := Rσ π(gσ −1 ). Let p : G → G/H and pσ : G → G/Hσ be the natural projections. Then the Poisson manifolds (G/H, p∗ πσ ) and (G/Hσ , (pσ )∗ π) are Poisson diffeomorphic. Proof. There is a natural diffeomorphism between G/H and G/Hσ given by ι : [g]H 7→ gσ −1 Hσ which satisfies p σ = ι ◦ p ◦ Rσ ,

where [g]H := gH ∈ G/H. We claim that ι is a Poisson map, i.e. ι∗ (p∗ πσ ) = (pσ )∗ π. Indeed ι∗ (p∗ πσ )

gσ −1

Hσ

= (Dι) |[g]H ((p∗ πσ ) ([g]H )) = (Dι) |[g]H ((Dp) |g (πσ (g)))

= (Dι) |[g]H (Dp) |g Rσ π(gσ −1 ) = D(ι ◦ p ◦ Rσ )|gσ−1 π(gσ −1 ) = D(pσ )|gσ−1 π(gσ −1 ) = ((pσ )∗ π) ( gσ −1 Hσ )

for any [g]H ∈ G/H and the claim follows.

2.3

Foliation point of view

It is somewhat unexpected that the π-coisotropy of a conjugate subgroup Adσ H is not related to the Adσ (π)-coisotropy of the subgroup H but related to the Rσ π-coisotropy of H. In this section, we use a foliation viewpoint to give a more conceptual explanation of this phenomenon. We call a foliation F on a manifold M regular if the leaf (i.e. the quotient) space M/F inherits a well-defined manifold structure from M . Let F be a regular foliation on a manifold M and ρ ∈ ∧k T M be a tensor field on M . We call F ρ-coisotropic if for any element [η] of the holonomy groupoid G that goes from s ∈ M to t ∈ M (and hence s, t belong to the same leaf L of F), there is a (leaf-preserving) local diffeomorphism η, implementing [η], from a neighborhood of s to a neighborhood of t with η (s) = t, such that (Dη)s (ρ (s)) − ρ (t) ∈ Tt L ∧ ∧k−1 Tt M , and hence ρ projects to a well-defined tensor field [ρ] = ρ/F on M/F. Note that the differential Dη of η is a local vector bundle map from T F to T F, where T F = ∪L∈F T L. 7

Fix a tensor field π on G. We consider the category C of G-manifolds M endowed with π-covariant tensor field ρ and a regular ρ-coisotropic foliation F that is invariant under the G-action on M . A morphism between two ˜ , ρ˜, F˜ is a smooth G-equivariant map φ : M → M ˜, objects (M, ρ, F) and M i.e. φ (gx) = gφ (x) for all (g, x) ∈ G×M , that induces a well-defined smooth ˜ /F˜ and sends the tensor field ρ on M to ρ˜ on M , i.e. map [φ] : M/F → M (Dφ) (ρ (x)) = ρ˜ (φ (x)) for all x ∈ M . It is easily recognized that the map [φ] induced by such a morphism φ is automatically G-equivariant and sends [ρ] to [˜ ρ]. It is natural to see that for a givenobject (M, ρ, F) of C, any diffeomorphism φ : M → M produces ˜ , ρ˜, F˜ of C with M ˜ = M , F˜ = φ∗ F whose leaves are exactly an object M the images of leaves of F under φ, and ρ˜ =(Dφ) (ρ), such that φ becomes ˜ = M, ρ˜, F˜ . In particular, an invertible morphism from (M, ρ, F) to M ρ˜ = (Dφ) (ρ) is π-covariant just like ρ, and F˜ = φ∗ F is ρ˜-coisotropic. For each connected closed subgroup H of G, the G-manifold G has a regular foliation FH with the right cosets gH, g ∈ G, as leaves, such that each holonomy groupoid element [η] is implemented by a right translation Rh with h ∈ H, which implies that for any tensor field ρ on G, FH is ρcoisotropic if and only if the subgroup H is ρ-coisotropic. Note that the diffeomorphism Rσ : G → G with σ ∈ G maps the foliation FH determined by H to the foliation FAdσ−1 H determined by Adσ−1 H, because it sends the leaf gH of FH to the leaf (gH) σ = gσ σ −1 Hσ = gσ Adσ−1 H of FAdσ−1 H for all g ∈ G. Thus Rσ determines an invertible morphism from (G, ρ, FH ) to G, Rσ ρ, FAdσ−1 H for any tensor field ρ on G that makes the subgroup H ρ-coisotropic. (This means that under the diffeomorphism Rσ , the tensor field ρ corresponds to Rσ ρ while the subgroup H corresponds to Adσ−1 H, not Rσ H which is not a subgroup.) In particular, Adσ−1 H is Rσ ρ-coisotropic when H is ρ-coisotropic. Since Rσ is invertible, we have Adσ−1 H is Rσ ρ-coisotropic if and only if H is ρ-coisotropic. Substituting H by Adσ H, we can also say that H is Rσ ρcoisotropic if and only if Adσ H is ρ-coisotropic. Furthermore from the above general discussion, it is also clear why the diffeomorphism Rσ induces a Poisson diffeomorphism (G/H = G/FH , ρ/FH ) → G/ Adσ−1 H = G/FAdσ−1 H , ρ/FAdσ−1 H . 8

3

Poisson Grassmannians

3.1

Coisotropic subgroups in standard SU(n)

Let us now restrict ourselves to the group SU (n) and fix the embedding of S(U (n − m) × U (m)) in SU (n) given by: A 0 (A, B) ֒→ . 0 B Recall that the standard Poisson–Lie tensor on SU (n) is defined, up to a constant factor by the Poisson 2–tensor π(g) = Lg r − Rg r where r ∈ g ∧ g, g = su(n) in the following, is the r-matrix given by X + − r= Xij ∧ Xij . 1≤i n − k the Poisson homogeneous spaces Xk,l are the non standard version of the Poisson homogeneous spaces denoted by U/KS0 in [Stok], where U = SU (n) and S, subset of the set of simple roots {α1 , . . . , αn−l−1 }, in su(n), is given by deleting αk−l . Such Poisson manifold should be compared with the standard Poisson quotient SU (n − l)/S(U (k − l) × U (n − k − l)) × 1l in the sense of understanding whether the two belong to a Poisson pencil, as it is the case for projective and Grassmann manifolds. 3. As a last remark let us consider the maximal torus T in SU (n) then π T = 0. This implies that T /(T ∩ Adσ(c,k) Kk ) is a family of 0– dimensional symplectic leaves in Grassmannians which can be explicitly described: T ∩ Adσ(c,k) Kk ≃ Tn−k ⇒

T ≃ Tk T ∩ Adσ(c,k) Kk

where the image of such points in the Grassmannian is given by √ √ (t1 , . . . , tk ) 7→ htj ( cen−j+1,n−j+1 − 1 − cej,n−j+1 ) j = 1, . . . , ki .

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