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

Abstract

Let $S$ be a closed connected oriented surface of genus $g \gt 0$. We study a Poisson subalgebra $W_1(g)$ of $C^{\infty}(\mathrm{Hom}(\pi_1(S), \mathrm{GL}(1, \mathbf{R}))/\mathrm{GL}(1, \mathbf{R}))$, the smooth functions on the moduli space of flat $\mathrm{GL}(1, \mathbf{R})$-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 $H^1(W_1(g), W_1(g) \wedge W_1(g)) \cong \mathrm{Hom}(\mathbf{Z}^{2g}, \mathbf{R})$. As a result, there is no cohomology class corresponding to the Turaev cobracket on $W_1(g)$.