Mapping class groups, skein algebras and combinatorial quantization

Abstract

The algebras $\mathcal{L}_{g,n}(H)$ have been introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the middle of the 1990's, in the program of combinatorial quantization of the moduli space of flat connections over the surface $\Sigma_{g,n}$ of genus $g$ with $n$ open disks removed. In this thesis we apply these algebras $\mathcal{L}_{g,n}(H)$ to low-dimensional topology (mapping class groups and skein algebras of surfaces), under the assumption that the gauge algebra $H$ is a finite dimensional factorizable ribbon Hopf algebra which is not necessarily semisimple, the guiding example being the restricted quantum group $\bar U_q(\mathfrak{sl}_2)$ (where $q$ is a $2p$-th root of unity). First, we construct from $\mathcal{L}_{g,n}(H)$ a projective representation of the mapping class group of $\Sigma_{g,0}$. We provide formulas for the representations of Dehn twists generating the mapping class group and we use these formulas to show that our representation is equivalent to the one constructed by Lyubashenko--Majid and Lyubashenko via categorical methods. For the torus $\Sigma_{1,0}$ with the gauge algebra $\bar U_q(\mathfrak{sl}_2)$, we compute explicitly the representation of $\mathrm{SL}_2(\mathbb{Z})$ and we determine its structure. Second, we introduce a diagrammatic description of $\mathcal{L}_{g,n}(H)$ which enables us to define in a very natural way the Wilson loop map $W$. This map associates an element of $\mathcal{L}_{g,n}(H)$ to any link in $(\Sigma_{g,n} \!\setminus\! D) \times [0,1]$ which is framed, oriented and colored by $H$-modules. When the gauge algebra is $H = \bar U_q(\mathfrak{sl}_2)$, we use $W$ and the representations of $\mathcal{L}_{g,n}(H)$ to construct representations of the skein algebras $\mathcal{S}_q(\Sigma_{g,n})$ for $q$ a $2p$-th root of unity. For the torus $\Sigma_{1,0}$ we explicitly study this representation.