Double Poisson algebras

TL;DR

This paper develops a non-commutative Poisson geometry framework that generalizes bi-symplectic geometry, enabling the study of Poisson structures on representation spaces and moduli spaces of certain algebras.

Contribution

It introduces a new non-commutative Poisson geometry theory that extends bi-symplectic geometry and applies it to moduli spaces of deformed multiplicative preprojective algebras.

Findings

Established Poisson structures on representation spaces.

Applied the theory to moduli spaces of deformed multiplicative preprojective algebras.

Connected non-commutative geometry with classical Poisson geometry.

Abstract

In this paper we develop Poisson geometry for non-commutative algebras. This generalizes the bi-symplectic geometry which was recently, and independently, introduced by Crawley-Boevey, Etingof and Ginzburg. Our (quasi-)Poisson brackets induce classical (quasi-)Poisson brackets on representation spaces. As an application we show that the moduli spaces of representations associated to the deformed multiplicative preprojective algebras recently introduced by Crawley-Boevey and Shaw carry a natural Poisson structure.