Lu, Jiang-Hua Classical dynamical \(r\)-matrices and homogeneous Poisson structures on \(G/H\) and \(K/T\). (English) Zbl 1008.53064 Commun. Math. Phys. 212, No. 2, 337-370 (2000). From author’s abstract: “Let \(G\) be a finite dimensional simple complex group equipped with the standard Poisson Lie group structure. We show that all \(G\)-homogeneous (holomorphic) Poisson structures on \(G/H\), where \(H\subset G\) is a Cartan subgroup, come from solutions to the classical dynamical Yang-Baxter equations which are classified by Etingof and Varchenko. A similar result holds for a maximal compact subgroup \(K\), and we get a family of \(K\)-homogeneous Poisson structures on \(K/T\), where \(T=K\cap H\) is a maximal torus of \(K\). This family exhausts all \(K\)-homogeneous Poisson structures on \(K/T\) up to isomorphisms. We study some Poisson geometrical properties of members of this family such as their symplectic leaves, their modular classes, and the moment maps for the \(T\)-action”. Reviewer: Raul Ibañez (Bilbão) Cited in 2 ReviewsCited in 15 Documents MSC: 53D17 Poisson manifolds; Poisson groupoids and algebroids 17B63 Poisson algebras 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 22E10 General properties and structure of complex Lie groups 37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) Keywords:classical dynamical \(r\)-matrices; Poisson structures; Yang-Baxter equation PDFBibTeX XMLCite \textit{J.-H. Lu}, Commun. Math. Phys. 212, No. 2, 337--370 (2000; Zbl 1008.53064) Full Text: DOI arXiv