Double Poisson Cohomology of Path Algebras of Quivers

In this note, we give a description of the graded Lie algebra of double derivations of a path algebra as a graded version of the necklace Lie algebra equipped with the Kontsevich bracket. Furthermore, we formally introduce the notion of double Poisson-Lichnerowicz cohomology for double Poisson algebras, and give some elementary properties. We introduce the notion of a linear double Poisson tensor on a quiver and show that it induces the structure of a finite dimensional algebra on the vector spaces V_v generated by the loops in the vertex v. We show that the Hochschild cohomology of the associative algebra can be recovered from the double Poisson cohomology. Then, we use the description of the graded necklace Lie algebra to determine the low-dimensional double Poisson-Lichnerowicz cohomology groups for three types of (linear and non-linear) double Poisson brackets on the free algebra in two variables. This allows us to develop some useful techniques for the computation of the double Poisson-Lichnerowicz cohomology.