Brackets in representation algebras of Hopf algebras

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.