A note on Poisson homogeneous spaces

TL;DR

This paper explores the structure of Poisson homogeneous spaces, identifying their cotangent bundle Lie algebroids, describing modular vector fields, and relating Poisson cohomology to relative Lie algebra cohomology, with applications to symplectic groupoids.

Contribution

It provides a new description of the cotangent bundle Lie algebroid of Poisson homogeneous spaces and links Poisson cohomology to relative Lie algebra cohomology, advancing understanding of their geometric structures.

Findings

Identified cotangent bundle Lie algebroid as a quotient of a transformation Lie algebroid.

Described modular vector fields of G/H.

Connected Poisson cohomology with relative Lie algebra cohomology.

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 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.