Relative invertibility of quantum groups at roots of unity

Location: Physics B21
Date(s): Wednesday 8th February 2023 (14:00-15:00)
Contact: Event Convenor Contact: Robert.Laugwitz@nottingham.ac.uk
Description: Speaker's Name: Patrick Kinnear
Speaker's Affiliation: University of Edinburgh
Speaker's Research Theme(s): Quantum mathematics,
Abstract:
Lusztig’s quantum group at a root of unity comes equipped with a quantum Frobenius map to the classical universal enveloping algebra, allowing us to pull back classical representations to representations of the quantum group. In fact, these pulled-back representations braid trivially with all others in Rep_q(G), and so they lie in the Muger centre and the functor Rep(G) → Rep_q(G) makes Rep_q(G) into a module category for Rep(G). This makes Rep_q(G) into an endomorphism of Rep(G) in the Morita theory SymTens of symmetric tensor categories. In this talk, we will discuss invertibility of this endomorphism. From the perspective of the Cobordism Hypothesis, this data defines a family of invertible TQFTs varying over the character variety (the moduli space of G-local systems) which extends the Crane-Yetter theory. In this picture, we expect to be able to define a non-semisimple version of Witten-Reshetikhin-Turaev theory, similar to the contemporary understanding of WRT as relative to CY.

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