Complete families of commuting functions for coisotropic Hamiltonian actions. (English) Zbl 1475.17039
The present paper deals with Hamiltonian actions of algebraic groups on affine manifolds focussing on the non-reductive case. As it is known, a Poisson algebra is a Lie algebra endowed with a associative product in such a way that the Lie and associative products are compatible via the Leibnitz identity. More precisely, in this paper, given an algebraic group \(G\) over a field \(\mathbb F\) of characteristic zero with \({\mathfrak g}= \mathrm{Lie}(G)\) then the dual space \(\mathfrak g^*\) is a Poisson variety and each irreducible \(G\)-subvariety \(X\subset\mathfrak g^*\) carries the induced Poisson structure. One proves that there is a set \(\{f_1, \cdots, f_l\}\subset \mathbb F[X]\) of algebraically independent polynomial functions, which are pairwise commuting with respect to the Poisson bracket, and such that \(l = (\dim X + \operatorname{tr.deg} \mathbb F(X)^G)/2)\). Finally several applications of this result are discussed to complete integrability of Hamiltonian systems on symplectic Hamiltonian \(G\)-varieties of corank zero and \(2\).
Reviewer: Béchir Dali (Bizerte)
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