Brackets in representation algebras of Hopf algebras
Gwenael Massuyeau, Vladimir Turaev

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 and , we define a commutative graded algebra representing the functor of -representations of . When is a cocommutative graded Hopf algebra and is a commutative ungraded Hopf algebra, we introduce a method deriving a Gerstenhaber bracket in from a Fox pairing in and a balanced biderivation in . 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.
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