Quasi-Poisson structures on representation spaces of surfaces

TL;DR

This paper introduces a canonical quasi-Poisson bracket on the space of N-dimensional representations of the fundamental group of a surface, extending known Poisson structures and utilizing a quasi-Poisson double algebra framework.

Contribution

It develops a new quasi-Poisson structure on representation spaces of surface groups, generalizing classical Poisson brackets and employing a novel algebraic approach.

Findings

Defined a canonical quasi-Poisson bracket for all N>0.

Extended the classical Poisson bracket on invariant functions.

Established a quasi-Poisson double algebra structure on the group algebra.

Abstract

Given an oriented surface S with base point * on the boundary, we introduce for all N>0, a canonical quasi-Poisson bracket on the space of N-dimensional linear representations of \pi_1(S,*). Our bracket extends the well-known Poisson bracket on GL_N-invariant functions on this space. Our main tool is a natural structure of a quasi-Poisson double algebra (in the sense of M. Van den Bergh) on the group algebra of \pi_1(S,*).