Double Poisson Structures on Finite Dimensional Semi-Simple Algebras

We give a description of the bimodule of double derivations 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 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 issn(S * T) of the free product S * T is given.