Covariant Poisson structures on complex Grassmannians

Abstract

The purpose of this paper is to study covariant Poisson structures on Gr n k 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]. * supported by PRIN Azioni di gruppi su varietá and GNSAGA.

Quantization of Poisson Manifolds from the Integrability of the Modular Function

“…Since U t (n) is coisotropic, u t (n) ⊥ is a Lie subalgebra and we will denote with U t (n) ⊥ ⊂ SB(n + 1, C) the subgroup integrating u t (n) ⊥ ⊂ sb(n + 1, C). An explicit expression for u t (n) ⊥ can be deduced from [7], page 16. As a consequence of coisotropy there is a uniquely defined Poisson structure π t on CP n = U t (n)\SU(n + 1) such that the quotient map p t : SU(n + 1) → CP n is Poisson.…”

“…It is relevant for what follows to remark here that as Poisson manifolds P k (t) do not depend on t (up to Poisson diffeomorphism); the dependence on t is manifested in the embeddings. What was not detailed in [7] is the way in which such Poisson submanifolds intersect along lower dimensional symplectic leaves. Let us remark the following.…”

“…Let us now move to the non standard case. Most of this description is contained in [7], to which we refer for unproven statements. By direct computation we have that if…”

Quantization of projective homogeneous spaces and duality principle

“…Then P = HK * is parabolic in G , H = P ∩ K and K H ≃ G P as smooth manifolds. It can be shown quite easily that the coisotropy of K implies the coisotropy of P , and furthermore, via Theorem 4.1 in [7], that K H and G P are also Poisson diffeomorphic. Thus in order to check whether P is coisotropic it is enough to check whether P ∩ K is coisotropic w.r. to the standard Poisson structure on the compact real group K .…”

“…Thus in order to check whether P is coisotropic it is enough to check whether P ∩ K is coisotropic w.r. to the standard Poisson structure on the compact real group K . There we can rely on results in [7], where a 1-parameter family of coisotropic subgroups H ε ⊆ SU(n) was given. Such subgroups induce a 1-parameter family of homogeneous Poisson quotients on complex Grassmannians.…”