We present several large classes of real Banach Lie-Poisson spaces whose characteristic distributions are integrable, the integral manifolds being symplectic leaves just as in finite dimensions. We also investigate when these leaves are embedded submanifolds or when they have Kähler structures. Our results apply to the real Banach Lie-Poisson spaces provided by the self-adjoint parts of predual...

The notion of a Poisson algebra was probably introduced in the first time by A.M. Vinogradov and J. S. Krasil’shchik 1975 under name “canonical algebra” Braconnier his short note “Algèbres de Poisson” (Comptes rendus Ac.Sci) 1977.

We compute the space of Poisson traces on symmetric powers of affine symplectic varieties. In the case of symplectic vector spaces, we also consider the quotient by the diagonal translation action, which includes the quotient singularities T C/Sn associated to the type A Weyl group Sn and its reflection representation C . We also compute the full structure of the natural D-module, previously de...

Let q be a Lie algebra over an algebraically closed field k of characteristic zero. The symmetric algebra S(q) has a natural structure of Poisson algebra, and our goal is to present a sufficient condition for the maximality of Poisson-commutative subalgebras of S(q) obtained by the argument shift method. Study of Poisson-commuttive subalgebras of S(q) has attracted much attention in the last ye...

In this paper we consider the Poisson algebraic structure associated with a classical r-matrix, i.e. with a solution of the modified classical Yang–Baxter equation. In Section 1 we recall the concept and basic facts of the r-matrix type Poisson orbits. Then we describe the r-matrix Poisson pencil (i.e the pair of compatible Poisson structures) of rank 1 or CPn-type orbits of SL(n,C). Here we ca...

The notions of vertex Lie algebra and vertex Poisson algebra are presented and connections among vertex Lie algebras, vertex Poisson algebras and vertex algebras are discussed.