Compatible Lie brackets related to elliptic curve

For the direct sum of several copies of sl_n, a family of Lie brackets compatible with the initial one is constructed. The structure constants of these brackets are expressed in terms of theta-functions associated with an elliptic curve. The structure of Casimir elements for these brackets is investigated. A generalization of this construction to the case of vector-valued theta-functions is presented. The brackets define a multi-hamiltonian structure for the elliptic sl_n-Gaudin model. A different procedure for constructing compatible Lie brackets based on the argument shift method for quadratic Poisson brackets is discussed.Comment: 18 pages, Late

Double Poisson brackets on free associative algebras

We discuss double Poisson structures in sense of M. Van den Bergh on free associative algebras focusing on the case of quadratic Poisson brackets. We establish their relations with an associative version of Young-Baxter equations, we study a bi-hamiltonian property of the linear-quadratic pencil of the double Poisson structure and propose a classification of the quadratic double Poisson brackets in the case of the algebra with two free generators. Many new examples of quadratic double Poisson brackets are proposed.Comment: 19 pages, late

Classification of integrable Vlasov-type equations

Classification of integrable Vlasov-type equations is reduced to a functional equation for a generating function. A general solution of this functional equation is found in terms of hypergeometric functions.Comment: latex, 15 pages, to appear in Theoretical and Mathematical Physic

Functional realization of some elliptic Hamiltonian structures and bosonization of the corresponding quantum algebras

We introduce a functional realization of the Hamiltonian structure on the moduli space of P-bundles on the elliptic curve E. Here P is parabolic subgroup in SL_n. We also introduce a construction of the corresponding quantum algebras.Comment: 20 pages, Amstex, minor change