Double Poisson brackets on free associative algebras

Abstract : We discuss double Poisson structures in sense of M. Van den Berghon free associative algebras focusing on the case of quadratic Poisson brackets.We establish their relations with an associative version of Yang-Baxter equa-tions, we study a bi-hamiltonian property of the linear-quadratic pencil of thedouble Poisson structure and propose a classification of the quadratic doublePoisson brackets in the case of the algebra with two free generators. Manynew examples of quadratic double Poisson brackets are proposed.