Poisson Brackets in Kontsevich's "Lie World"

TL;DR

This paper develops the concept of Poisson brackets within Kontsevich's 'Lie World', establishing their uniqueness, and applies this to linearize structures on moduli spaces, linking quasi-Poisson and Hamiltonian spaces.

Contribution

It introduces a universal definition of Poisson brackets in Kontsevich's framework and proves their uniqueness, with applications to moduli space structures and equivalences between Hamiltonian spaces.

Findings

Uniqueness of Poisson brackets with a given moment map.

Linearization of quasi-Poisson structures on moduli spaces.

Identification of symplectic leaves with coadjoint orbit reductions.

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.