  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.
