Cell 2-representations and categorification at prime roots of unity

Robert Laugwitz, Vanessa Miemietz

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Abstract

Motivated by recent advances in the categorification of quantum groups at prime roots of unity, we develop a theory of 2-representations for 2- categories, enriched with a p-differential, which satisfy finiteness conditions analogous to those of finitary or fiat 2-categories. We construct cell 2-representations in this setup, and consider a class of 2-categories stemming from bimodules over a p-dg category in detail. This class is of particular importance in the categorification of quantum groups, which allows us to apply our results to cyclotomic quotients of the categorifications of small quantum group of type sl2 at prime roots of unity by Elias–Qi [Advances in Mathematics 288 (2016)]. Passing to stable 2-representations gives a way to construct triangulated 2-representations, but our main focus is on working with p-dg enriched 2-representations that should be seen as a p-dg enhancement of these triangulated ones.
Original languageEnglish
Article number106937
Number of pages66
JournalAdvances in Mathematics
Volume361
Early online date31 Dec 2019
DOIs
Publication statusPublished - 12 Feb 2020

Keywords

  • 2-Representation theory
  • Categorification at roots of unity
  • Enriched 2-categories
  • Hopfological algebra
  • QUANTUM-FIELD THEORY
  • SIMPLE TRANSITIVE 2-REPRESENTATIONS

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