On linear-quadratic Poisson pencils on trivial central extensions of semisimple Lie algebras

TL;DR

This paper develops the theory of quadratic Poisson structures compatible with linear structures on central extensions of semisimple Lie algebras, revealing new families of compatible structures and their relation to integrable systems.

Contribution

It introduces a general theory for quadratic Poisson structures on trivial central extensions and constructs a 10-parametric family on (3)*, linking to known integrable systems.

Findings

Existence of a 10-parametric family of quadratic Poisson structures on (3)*

Construction of involutive families of functions including elliptic Calogero--Moser Hamiltonian

Compatibility of new structures with canonical linear Poisson structures

Abstract

The paper is devoted to quadratic Poisson structures compatible with the canonical linear Poisson structures on trivial 1-dimensional central extensions of semisimple Lie algebras. In particular, we develop the general theory of such structures and study related families of functions in involution. We also show that there exists a 10-parametric family of quadratic Poisson structures on $\gl(3)^*$ compatible with the canonical linear Poisson structure and containing the 3-parametric family of quadratic bivectors recently introduced by Vladimir Sokolov. The involutive family of polynomial functions related to the corresponding Poisson pencils contains the hamiltonian of the polynomial form of the elliptic Calogero--Moser system.