Let Σ g,n be a compact oriented surface of genus g with n open disks removed. The graph algebra 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 Σ g,n . We construct a projective representation of the mapping class group of Σ g,n using 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. their regular support and their useful remarks. I thank A. Gainutdinov for inviting me to present this work at the algebra seminar of the University of Hamburg.
Notations.If A is a C-algebra, V is a finite-dimensional A-module and x ∈ A, we denote by V x ∈ End C (V ) the representation of x on the module V . Similarly, if X ∈ A ⊗n and if V 1 , . . . , V n are A-modules, we denote by V 1 ...Vn X the representation of X on V 1 ⊗ . . . ⊗ V n . Here we consider only finite-dimensional representations, hence H-module implicitly means finite-dimensional H-module. IJ R 12 I T 1 J T 2 = J T 2 I T 1 IJ R 12 . Since H is finite-dimensional, it exists right and left integrals µ r , µ l ∈ O(H) defined by I B or I M , let J V be J A or J B or J M and let i < j. Then, by definition of the right action and by (18):