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 = Uq(sl(2)). We also show that it is equivalent to the representation constructed by Lyubashenko and Majid.