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

Symmetry Integrability and Geometry-Methods and Applications

article

Matthieu Faitg¹ 

[1] IMAG, Univ Montpellier, CNRS

关键词: combinatorial quantization;    factorizable Hopf algebra;    modular group;    restricted quantum group;   

DOI  :  10.3842/SIGMA.2019.077

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown