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

DOI10.2996/KMJ/1540951256zbMath1432.57040arXiv1710.03478OpenAlexW2963015992MaRDI QIDQ1756012

Moeka Nobuta

Publication date: 11 January 2019

Published in: Kodai Mathematical Journal (Search for Journal in Brave)

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 $W_1(g)$.

Full work available at URL: https://arxiv.org/abs/1710.03478

zbMATH Keywords

Wilson loopcobracketflat bundleGoldman Lie algebra

Mathematics Subject Classification ID

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

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