Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 077, 39 pages      arXiv:1805.00924      https://doi.org/10.3842/SIGMA.2019.077

Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras

Matthieu Faitg
IMAG, Univ Montpellier, CNRS, Montpellier, France

Received February 02, 2019, in final form September 24, 2019; Published online October 03, 2019

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.

Key words: combinatorial quantization; factorizable Hopf algebra; modular group; restricted quantum group.

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