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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)
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