Brackets in representation algebras of Hopf algebras

TL;DR

This paper constructs a Gerstenhaber bracket on representation algebras of Hopf algebras, generalizing classical Poisson structures on moduli spaces, inspired by non-commutative geometry.

Contribution

It introduces a new method to derive Gerstenhaber brackets in representation algebras of Hopf algebras using Fox pairings and biderivations, extending non-commutative Poisson geometry.

Findings

Defined a commutative graded algebra representing B-representations of A

Derived a Gerstenhaber bracket from Fox pairings and biderivations

Connected algebraic structures to classical Poisson geometries

Abstract

For any graded bialgebras $A$ and $B$, we define a commutative graded algebra $A_B$ representing the functor of $B$-representations of $A$. When $A$ is a cocommutative graded Hopf algebra and $B$ is a commutative ungraded Hopf algebra, we introduce a method deriving a Gerstenhaber bracket in $A_B$ from a Fox pairing in $A$ and a balanced biderivation in $B$. Our construction is inspired by Van den Bergh's non-commutative Poisson geometry, and may be viewed as an algebraic generalization of the Atiyah--Bott--Goldman Poisson structures on moduli spaces of representations of surface groups.