# Brackets in representation algebras of Hopf algebras

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.
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Journal articles
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https://hal.archives-ouvertes.fr/hal-01189019
Contributor : Gwénaël Massuyeau <>
Submitted on : Tuesday, September 1, 2015 - 9:48:56 AM
Last modification on : Thursday, August 1, 2019 - 3:18:19 PM

### Citation

Gwenael Massuyeau, Vladimir Turaev. Brackets in representation algebras of Hopf algebras. Journal of Noncommutative Geometry, European Mathematical Society, 2018, 12 (2), pp.577-636. ⟨10.4171/JNCG/286⟩. ⟨hal-01189019⟩

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