Poisson Brackets in Kontsevich's "Lie World"

Abstract

In this note the notion of Poisson brackets in Kontsevich's "Lie World" is developed. These brackets can be thought of as "universally" defined classical Poisson structures, namely formal expressions only involving the structure maps of a quadratic Lie algebra. We prove a uniqueness statement about these Poisson brackets with a given moment map. As an application we get formulae for the linearization of the quasi-Poisson structure of the moduli space of flat connections on a punctured sphere, and thereby identify their symplectic leaves with the reduction of coadjoint orbits. Equivalently, we get linearizations for the Goldman double Poisson bracket, our definition of Poisson brackets coincides with that of Van Den Bergh in this case. This can furthermore be interpreted as giving a monoidal equivalence between Hamiltonian quasi-Poisson spaces and Hamiltonian spaces.