The algebra L g,n (H) was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and quantizes the character variety of the Riemann surface Σ g,n \ D (D is an open disk). In this article we define a holonomy map in that quantized setting, which associates a tensor with components in L g,n (H) to tangles in (Σ g,n \D) × [0, 1], generalizing previous works of Buffenoir-Roche and Bullock-Frohman-Kania-Bartoszynska. We show that holonomy behaves well for the stack product and the action of the mapping class group; then we specialize this notion to links in order to define a generalized Wilson loop map. Thanks to the holonomy map, we give a geometric interpretation of the vacuum representation of L g,0 (H) on L 0,g (H). Finally, the general results are applied to the case H = U q 2 (sl 2 ) in relation to skein theory and the most important consequence is that the stated skein algebra of a compact oriented surface with just one boundary edge is isomorphic to L g,n U q 2 (sl 2 ) . Throughout the paper we use a graphical calculus for tensors with coefficients in L g,n (H) which makes the computations and definitions very intuitive.

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):

Let Σ g,n be a compact oriented surface of genus g with n open disks removed. The 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 . Here we focus on the two building blocks L 0,1 (H) and L 1,0 (H) under the assumption that the gauge Hopf algebra H is finite-dimensional, factorizable and ribbon, but not necessarily semi-simple. We construct a projective representation of SL 2 (Z), the mapping class group of the torus, using L 1,0 (H) and we study it explicitly for H = U q (sl (2)). We also show that it is equivalent to the representation constructed by Lyubashenko and Majid.and this give the result sinceIt follows that ψ ∈ H * is symmetric if, and only if, J C (±) ⊲ ψ = ψI dim(J) for all J.

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