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October 2018 On cobrackets on the Wilson loops associated with flat $\mathrm{GL}(1, \mathbf{R})$-bundles over surfaces
Moeka Nobuta
Kodai Math. J. 41(3): 591-619 (October 2018). DOI: 10.2996/kmj/1540951256

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 $W_1(g)$.

Citation

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Moeka Nobuta. "On cobrackets on the Wilson loops associated with flat $\mathrm{GL}(1, \mathbf{R})$-bundles over surfaces." Kodai Math. J. 41 (3) 591 - 619, October 2018. https://doi.org/10.2996/kmj/1540951256

Information

Published: October 2018
First available in Project Euclid: 31 October 2018

zbMATH: 07000586
MathSciNet: MR3870706
Digital Object Identifier: 10.2996/kmj/1540951256

Rights: Copyright © 2018 Tokyo Institute of Technology, Department of Mathematics

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Vol.41 • No. 3 • October 2018
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