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 $\Sigma _{g,n}$. We construct a projective representation of the mapping class group of $\Sigma _{g,n}$ using $\mathcal {L}_{g,n}(H)$ and its subalgebra of invariant elements. Here we assume that the gauge Hopf algebra H is finite-dimensional, factorizable and ribbon, but not necessarily semi-simple. We also give explicit formulas for the representation of the Dehn twists generating the mapping class group; in particular, we show that it is equivalent to a representation constructed by V. Lyubashenko using categorical methods.