Classical dynamical r-matrices, Poisson homogeneous spaces, and Lagrangian subalgebras

TL;DR

This paper explores conditions under which dynamical r-matrices uniquely determine Poisson homogeneous structures on quotient spaces, extending Lu's correspondence and classifying structures for complex simple groups.

Contribution

It establishes general conditions for the Lu correspondence to be one-to-one and applies these to classify triangular Poisson structures on certain homogeneous spaces.

Findings

Identified conditions ensuring the Lu correspondence is bijective.

Classified all triangular Poisson homogeneous structures for simple complex groups.

Extended the understanding of the relationship between r-matrices and Poisson structures.

Abstract

Jiang-Hua Lu showed that any dynamical r-matrix for the pair $(g,u)$ naturally induces a Poisson homogeneous structure on $G/U$. She also proved that if $g$ is complex simple, $u$ is its Cartan subalgebra and $r$ is quasitriangular, then this correspondence is in fact 1-1. In the present paper we find some general conditions under which the Lu correspondence is 1-1. Then we apply this result to describe all triangular Poisson homogeneous structures on $G/U$ for a simple complex group $G$ and its reductive subgroup $U$ containing a Cartan subgroup.