Compatible Lie brackets related to elliptic curve

Abstract

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.