Let $g$ be a finite dimensional Lie algebra, and let me denote $A=(\bigwedge g^* \otimes g, d)$ the Chevalley-Eilenberg complex that calculates cohomology of the Lie algebra with coefficients in the adjoint representation. On the complex $A$ there is Nijenhuis–Richardson bracket and the resulting DGLA could be used to describe deformations of the given algebra.
Suppose that $g$ is unimodular i.e. determinant of the adjoint representation is trivial representation. In such situation there are isomorphisms $$ \bigwedge^i g^* \otimes g \cong \bigwedge^{n-i} g \otimes g, $$ thus differential in the complex for homology of the lie algebra induces one more map on the complex (this map goes in the opposite direction on $A$). I want to mimic Ginzburg's construction of the Batalin-Vilkovisky algebra structure on the Hochshild cohomology complex of a CY algebra and such induced map should play a role of BV differential, and unimodularity condition plays role of the CY condition. But one important ingredient is missing in this situation, namely there is no associative algebra structure on $A$.
I wounder is it possible to introduce some multiplication on $A$ such that Nijenhuis–Richardson bracket become a BV bracket?
