On double brackets for marked surfaces

TL;DR

This paper introduces a noncommutative generalization of the Goldman bracket for marked surfaces, constructing a double quasi-Poisson bracket on the group algebra of the twisted fundamental group, with applications to local systems.

Contribution

It proposes a novel double quasi-Poisson bracket construction on the group algebra of the twisted fundamental group of a marked surface, extending Goldman bracket concepts.

Findings

Constructed a double quasi-Poisson bracket on the group algebra of the twisted fundamental group.

Showed the bracket generalizes the Goldman bracket to a noncommutative setting.

Applied the construction to local systems on surfaces with various structure groups.

Abstract

We propose a construction of a double quasi-Poisson bracket on the group algebra associated to the twisted fundamental group of a marked oriented surface $(S,P)$ with boundary, where $P$ is a finite set of marked points on the boundary of the surface $S$ such that on every boundary component there is at least one point of $P$. We show that this double bracket is a noncommutative generalization of the well-known Goldman bracket, defined on the space of free homotopy classes of loops on $S$. For an algebra $A$ without polynomial identities, we construct a double bracket on the space of decorated twisted $\mathrm{GL}_n(A)$-, symplectic and indefinite orthogonal local systems.