Invariant Poisson Structures on Homogeneous Manifolds: Algebraic Characterization, Symplectic Foliation, and Contravariant Connections

Abstract

In this paper, we study invariant Poisson structures on homogeneous manifolds, which serve as a natural generalization of homogeneous symplectic manifolds previously explored in the literature. Our work begins by providing an algebraic characterization of invariant Poisson structures on homogeneous manifolds. More precisely, we establish a connection between these structures and solutions to a specific type of classical Yang-Baxter equation. This leads us to explain a bijective correspondence between invariant Poisson tensors and class of Lie subalgebras: For a connected Lie group $G$ with lie algebra $\mathfrak{g}$, and $H$ a connected closed subgroup with Lie algebra $\mathfrak{h}$, we demonstrate that the class of $G$-invariant Poisson tensors on $G/H$ is in bijective correspondence with the class of Lie subalgebras $\mathfrak{a} \subset \mathfrak{g}$ containing $\mathfrak{h}$, equipped with a $2$-cocycle $\omega$ satisfying $\mathrm{Rad}(\omega) = \mathfrak{h}$. Then, we explore numerous examples of invariant Poisson structures, focusing on reductive and symmetric pairs. Furthermore, we show that the symplectic foliation associated with invariant Poisson structures consists of homogeneous symplectic manifolds. Finally, we investigate invariant contravariant connections on homogeneous spaces endowed with invariant Poisson structures. This analysis extends the study by K. Nomizu of invariant covariant connections on homogeneous spaces.