2015/11/09 by Jiang-Hua Lu, Lu, Jiang-Hua, Victor Mouquin +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1511.02559
openalex publication_date 2015/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a connected abelian Lie group T acting on a Poisson manifold (Y,π) by Poisson isomorphisms, the T-leaves of π in Y are, by definition, the orbits of the symplectic leaves of π under T, and the leaf stabilizer of a T-leaf is the subspace of the Lie algebra of T that is everywhere tangent to all the symplectic leaves in the T-leaf. In this paper, we first develop a general theory on T-leaves and leaf stabilizers for a class of Poisson structures defined by Lie bialgebra actions and quasitriangular r-matrices. We then apply the general theory to four series of holomorphic Poisson structures on products of flag varieties and related spaces of a complex semi-simple Lie group G. We describe their T-leaf decompositions, where T is a maximal torus of G, in terms of (open) extended Richardson varieties and extended double Bruhat cells associated to conjugacy classes of G, and we compute their leaf stabilizers and the dimension of the symplectic leaves in each T -leaf.