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

TL;DR

This paper constructs a projective representation of the modular group for non-semisimple quantum groups using combinatorial quantization, extending known theories to more general algebraic structures.

Contribution

It develops a new projective representation of the mapping class group for non-semisimple Hopf algebras, linking it to existing Lyubashenko-Majid representations.

Findings

Constructed a projective SL(2,Z) representation from $ ext{L}_{1,0}(H)$

Explicit analysis for $H = ar{U}_q(sl(2))$

Proved equivalence with Lyubashenko-Majid representation

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.