Graded Necklace Lie Bialgebras and Batalin-Vilkovisky Formalism

TL;DR

This paper introduces a graded generalization of necklace Lie bialgebras associated with quivers and explores their Batalin-Vilkovisky structures, connecting algebraic and geometric perspectives through twisted trace morphisms.

Contribution

It develops a graded extension of necklace Lie bialgebras depending on quivers and relates their BV structures to symplectic forms on representation varieties.

Findings

Established a graded necklace Lie bialgebra framework.

Connected BV structures from algebraic and geometric viewpoints.

Defined a twisted trace morphism linking different BV algebras.

Abstract

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.