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Classical dynamical \(r\)-matrices and homogeneous Poisson structures on \(G/H\) and \(K/T\). (English) Zbl 1008.53064

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”.

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)
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