Holonomy and (stated) skein algebras in combinatorial quantization

Abstract

The algebra ${\mathcal{L}}_{g,n}(H)$ was introduced by Alekseev, Grosse, and Schomerus and by Buffe­noir and Roche and quantizes the character variety of the Riemann surface ${\Sigma }_{g,n}\setminus D$ (where $D$ is an open disk). In this article we define a holonomy map in that quantized setting, which associates a tensor with components in ${\mathcal{L}}_{g,n}(H)$ to tangles in $({\Sigma }_{g,n}\setminus D)\times [0,1]$, generalizing previous works of Buffenoir and Roche and of Bullock, Frohman, and 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 ${\mathcal{L}}_{g,0}(H)$ on ${\mathcal{L}}_{0,g}(H)$. Finally, the general results are applied to the case $H=U_{q^2}({\mathfrak{sl}}_2)$ 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 ${\mathcal{L}}_{g,n}(U_{q^2}({\mathfrak{sl}}_2))$. Throughout the paper, we use a graphical calculus for tensors with coefficients in ${\mathcal{L}}_{g,n}(H)$ which makes the computations and definitions very intuitive.