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

The paper is devoted to quadratic Poisson structures compatible with the canonical linear Poisson structures on (necessarily) 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 introduced in 2017 by Vladimir Sokolov, who showed that the corresponding involutive family of functions contains the hamiltonian of the polynomial form of the elliptic Calogero–Moser system. We also explicitly write the normal forms of the Poisson pencils in the 10-parametric family and related integrable systems. They correspond to normal forms of ternary cubic forms (degenerations of normal elliptic curve in P2).