A note on Poisson homogeneous spaces

Abstract

Abstract. We identify the cotangent bundle Lie algebroid of a Poisson homogeneous space G/H of a Poisson Lie group G as a quotient of a transformation Lie algebroid over G. As applications, we describe the modular vector fields of G/H, and we identify the Poisson cohomology of G/H with coefficients in powers of its canonical line bundle with relative Lie algebra cohomology of the Drinfeld Lie algebra associated to G/H. We also construct a Poisson groupoid over (G/H, π) which is symplectic near the identity section. This note serves as preparation for forthcoming papers, in which we will compute explicitly the Poisson cohomology and study their symplectic groupoids for certain examples of Poisson homogeneous spaces related to semi-simple Lie groups.

The quantization of the symplectic groupoid of the standard Podle s ̀ sphere

“…i.e. the spectrum of ρ(τ ) given in (15) for q = e − /2 . This formula differs by a factor from (31) obtained by means of a complex polarization.…”

“…The first step of the program is to construct an explicit description of the symplectic groupoid integrating the semiclassical sphere and to discuss the modular function as the function that integrates the modular vector field. We will use a description of the integration of Poisson homogeneous spaces that has been given only recently in [1,15] (see also [23,9]). Relying on the explicit computation of Poisson cohomology given in [10], we can conclude that the prequantization cocycle is non trivial.…”

Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

“…In [126], Ping Xu determines how the modular class of a Dirac submanifold of a Poisson manifold is related to that of the ambient Poisson manifold. J.-H. Lu recently described the modular classes of Poisson homogeneous spaces [91], generalizing a result of [34].…”

AdS Poisson homogeneous spaces and Drinfel’d doubles

“…At this point, it is worth recalling that in the same manner as quantum groups can be thought of as Hopf algebra counterparts of Poisson-Lie groups, quantum homogeneous spaces can be understood as quantisations of Poisson homogeneous spaces, for an overview see [29,[36][37][38][39][40][41][42][43][44][45] and references therein. While much simpler on a computational level, these Poisson homogeneous spaces still carry relevant information about the associated quantum homogeneous spaces and can be viewed as their semiclassical limits.…”