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

Let Sigma(g,n) be a compact oriented surface of genus g with n open disks removed. The algebra L-g,L-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 L-0,L-1(H) and L-1,L-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 L-1,L-0(H) and we study it explicitly for H = (U) over bar (q)(sl(2)). We also show that it is equivalent to the representation constructed by Lyubashenko and Majid.