Research Seminar in Mathematics - On quantum representations of mapping class groups from a finite group

Speaker

Jens Fjelstad, Örebro University.

Abstract

Topological quantum field theory (TQFT) is a gadget that produces two kinds of data, topological invariants and finite dimensional representations of mapping class groups of surfaces, so called quantum representations. These data are of interest both for Mathematics and Physics, one recent application being in quantum computing. One construction of TQFT is via a certain Hopf algebra constructed from a finite group, producing the quantum representations relevant for this talk. I will briefly introduce the ingredients such as mapping class groups of surfaces and the class of quantum representations associated to a finite group, and then discuss work (partly in collaboration with Jürgen Fuchs) to determine their structure in terms of finite group data. Recent results are focused on the restriction of a quantum representation to the terms in a filtration of a mapping class group known as the Johnson filtration.