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Nonsemisimple quantum invariants and TQFTs from small and unrolled quantum groups

Marco De Renzi, Nathan Geer and Bertrand Patureau-Mirand

Algebraic & Geometric Topology 20 (2020) 3377–3422
Abstract

We show that unrolled quantum groups at odd roots of unity give rise to relative modular categories. These are the main building blocks for the construction of (1+1+1)–TQFTs extending CGP invariants, which are nonsemisimple quantum invariants of closed 3–manifolds decorated with ribbon graphs and cohomology classes. When we consider the zero cohomology class, these quantum invariants are shown to coincide with the renormalized Hennings invariants coming from the corresponding small quantum groups.

Keywords
nonsemisimple quantum invariants, nonsemisimple TQFTs, small quantum groups, unrolled quantum groups
Mathematical Subject Classification 2010
Primary: 57M27, 81R50
References
Publication
Received: 1 January 2019
Revised: 11 December 2019
Accepted: 3 January 2020
Published: 29 December 2020
Authors
Marco De Renzi
Department of Mathematics
Waseda University
Tokyo
Japan
Nathan Geer
Department of Mathematics and Statistics
Utah State University
Logan, UT
United States
Bertrand Patureau-Mirand
UMR 6205, LMBA
Université de Bretagne-Sud
Vannes
France