Double Poisson structures on finite dimensional semi-simple algebras

We give a description of the bimodule of double derivations DDer(S) of a finite dimensional semi-simple algebra S and its double Schouten bracket in terms of a quiver. This description is used to determine which degree two monomials in T_SDDer(S) induce double Poisson brackets on S. In case S = C^n, a criterion for any degree two element to give a double Poisson bracket is deduced. For S = C^n and S' = C^m, the induced Poisson bracket on the variety of isomorphism classes of semi-simple representations iss_n(S*T) of the free product S*T is given.