## On cobrackets on the Wilson loops associated with flat $$\mathrm{GL}(1,\mathbb{R})$$-bundles over surfaces.(English)Zbl 1432.57040

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

### MSC:

 57M05 Fundamental group, presentations, free differential calculus 51H20 Topological geometries on manifolds 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B62 Lie bialgebras; Lie coalgebras 57K20 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)

### Keywords:

cobracket; flat bundle; Wilson loop; Goldman Lie algebra
Full Text: