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An involutive Lie bialgebra induces a Batalin-Vilkovisky operator on its exterior algebra. We introduce a graded generalization of the necklace Lie bialgebra, which depends on a choice of a quiver Q. We relate the resulting Batalin-Vilkovisky structure to the Batalin-Vilkovisky structure coming from a degree -1 symplectic form on a suitably defined representation variety of the quiver Q. The morphism intertwining these Batalin-Vilkovisky algebras will be given by a twisted trace, recovering the usual (super) trace and the odd trace.
Perry et al. (Fri,) studied this question.