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This article is cited in 2 scientific papers (total in 2 papers)
Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras
Matthieu Faitg IMAG, Univ Montpellier, CNRS, Montpellier, France
Abstract:
Let $\Sigma_{g,n}$ be a compact oriented surface of genus $g$ with $n$ open disks removed. The algebra $\mathcal{L}_{g,n}(H)$ was introduced by Alekseev–Grosse–Schomerus and Buffenoir–Roche and is a combinatorial quantization of the moduli space of flat connections on $\Sigma_{g,n}$. Here we focus on the two building blocks $\mathcal{L}_{0,1}(H)$ and $\mathcal{L}_{1,0}(H)$ under the assumption that the gauge Hopf algebra $H$ is finite-dimensional, factorizable and ribbon, but not necessarily semisimple. We construct a projective representation of $\mathrm{SL}_2(\mathbb{Z})$, the mapping class group of the torus, based on $\mathcal{L}_{1,0}(H)$ and we study it explicitly for $H = \overline{U}_q(\mathfrak{sl}(2))$. We also show that it is equivalent to the representation constructed by Lyubashenko and Majid.
Keywords:
combinatorial quantization, factorizable Hopf algebra, modular group, restricted quantum group.
Received: February 2, 2019; in final form September 24, 2019; Published online October 3, 2019
Citation:
Matthieu Faitg, “Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras”, SIGMA, 15 (2019), 077, 39 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1513 https://www.mathnet.ru/eng/sigma/v15/p77
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Abstract page: | 94 | Full-text PDF : | 42 | References: | 5 |
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