  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.
