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

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.