Projective representations of mapping class groups in combinatorial quantization

Abstract

Let ${\Sigma }_{g,n}$ be a compact oriented surface of genus $g$ with $n$ open disks removed. The graph 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.

32 pages; minor corrections and improvements; new section and new theorem added

References in corpus (1)

• Integrating quantum groups over surfaces

Cited by in corpus (1)

• Finite presentations for stated skein algebras and lattice gauge field theory