  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.
