Poisson brackets in Kontsevich’s “Lie World”

Abstract

In this note we develop the theory of double brackets in the sense of van den Bergh (2008) in Kontsevich’s non-commutative “Lie World”. These double brackets can be thought of as Poisson structures defined by formal expressions only involving the structure maps of a quadratic Lie algebra. The basic example is the Kirillov–Kostant–Souriau (KKS) Poisson bracket.We introduce a notion of non-degenerate double brackets. Surprisingly, in this framework the KKS bracket turns out to be non-degenerate. The main result of the paper is the uniqueness theorem for double brackets with a given moment map. As applications, we establish a monoidal equivalence between Hamiltonian quasi-Poisson spaces and Hamiltonian spaces and give a new proof of the theorem by L. Jeffrey in Jeffrey (1994) on symplectic structure on the moduli space of flat g-connections on a surface of genus 0.