On cobrackets on the Wilson loops associated with flat $\mathrm{GL}(1, \mathbb{R})$-bundles over surfaces

TL;DR

This paper investigates the algebraic structures of functions on the moduli space of flat L(1, \u211d) bundles over surfaces, classifies cobrackets, and shows the Turaev cobracket does not appear in this context.

Contribution

It classifies all cobrackets on the Poisson subalgebra of functions on the moduli space, revealing the absence of the Turaev cobracket in this setting.

Findings

Classified all cobrackets on the algebra up to coboundary.

Computed the first cohomology group of the algebra.

Established the non-existence of the Turaev cobracket in this framework.

Abstract

Let $S$ be a closed connected oriented surface of genus $g>0$. We study a Poisson subalgebra $W_1(g)$ of $C^{\infty}(\mathrm{Hom}(\pi_1(S), \mathrm{GL}(1, \mathbb{R}))/\mathrm{GL}(1, \mathbb{R}))$, the smooth functions on the moduli space of flat $\mathrm{GL}(1, \mathbb{R})$-bundles over $S$. There is a surjective Lie algebra homomorphism from the Goldman Lie algebra onto $W_1(g)$. We classify all cobrackets on $W_1(g)$ up to coboundary, that is, we compute $H^1(W_1(g), W_1(g)\wedge W_1(g))\cong \mathrm{Hom}(\mathbb{Z}^{2g}, \mathbb{R})$. As a result, there is no cohomology class corresponding to the Turaev cobracket on $W_1(g)$.