Quasi-Poisson Structures on Representation Spaces of Surfaces

Abstract

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