Complete families of commuting functions for coisotropic Hamiltonian actions

2005/11/20 by É. B. Vinberg, Vinberg, E. B., Oksana Yakimova +1
Mathematics · #17B63 #53D17 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.math/0511498

openalex publication_date 2005/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be an algebraic group over a field F of characteristic zero, with Lie algebra g=Lie(G). The dual space g^* equipped with the Kirillov bracket is a Poisson variety and each irreducible G-invariant subvariety X⊂ g^* carries the induced Poisson structure. We prove that there is a family of algebraically independent polynomial functions f₁,...f_l on X, which pairwise commute with respect to the Poisson bracket and such that l=(dim X+tr.deg F(X)^G)/2. We also discuss several applications of this result to complete integrability of Hamiltonian systems on symplectic Hamiltonian G-varieties of corank zero and 2.

Citations

• The centralisers of nilpotent elements in classical Lie algebras

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