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

Abstract

Let Σg,n be a compact oriented surface of genus g with n open disks removed. The algebra Lg,n(H) was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and is a combinatorial quantization of the moduli space of flat connections on Σg,n. Here we focus on the two building blocks L₀,₁(H) and L₁,₀(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 SL₂(Z), the mapping class group of the torus, based on L₁,₀(H), and we study it explicitly for H = Ūq(sl(2)). We also show that it is equivalent to the representation constructed by Lyubashenko and Majid.