Invariant Poisson Structures on Homogeneous Manifolds: Algebraic Characterization, Symplectic Foliation, and Contravariant Connections
Abdelhak Abouqateb, Charif Bourzik
TL;DR
This work provides a complete algebraic characterization of $G$-invariant Poisson structures on homogeneous spaces $G/H$ via $r$-matrices solving a Yang–Baxter type equation $\llbracket r,r\rrbracket=0$, yielding a bijection with Lie data $(\mathfrak{a},\omega)$ where $\mathfrak{h}\subset \mathfrak{a}\subset \mathfrak{g}$ and $\omega$ is a $2$-cocycle with $\mathrm{Rad}(\omega)=\mathfrak{h}$. It specializes cleanly to reductive and symmetric pairs, linking invariant bivectors to $r$ in $\wedge^2 \mathfrak{m}$ and providing concrete computations in examples such as $\mathrm{GL}_n^+\!(\mathbb{R})$ acting on $\mathcal{S}_n^{++}(\mathbb{R})$ and $\mathrm{SO}_4(\mathbb{R})$ on $\mathrm{G}^{+}_{2}(\mathbb{R}^4)$. The paper also characterizes the regular symplectic foliation as a family of homogeneous symplectic leaves $A_r/H$ and describes the induced leaf-space geometry. Finally, it develops a robust framework for invariant contravariant connections on $(G/H,\pi)$, showing a one-to-one correspondence with Ad$(H)$-invariant bilinear maps $\mathfrak{b}: \mathfrak{m}^*\times\mathfrak{m}^*\to\mathfrak{m}^*$ and relating these to covariant data via a precise formula for the connection and its curvature and torsion, including Fedosov-type variants.
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 $ω$ satisfying $\mathrm{Rad}(ω) = \mathfrak{h}$. 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.