Cell 2-Representations and Categorification at Prime Roots of Unity

Laugwitz, Robert; Miemietz, Vanessa

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 _at 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.

Citation

Laugwitz, R., & Miemietz, V. (2020). Cell 2-Representations and Categorification at Prime Roots of Unity. Advances in Mathematics, 361, https://doi.org/10.1016/j.aim.2019.106937

• Journal Article Type: Article
• Acceptance Date: Nov 28, 2019
• Online Publication Date: Dec 31, 2019
• Publication Date: Feb 12, 2020
• Deposit Date: Dec 19, 2019
• Publicly Available Date: Jan 1, 2021
• Journal: Advances in Mathematics
• Print ISSN: 0001-8708
• Electronic ISSN: 1090-2082
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 361
• Article Number: 106937
• DOI: https://doi.org/10.1016/j.aim.2019.106937
• Keywords: 2-representation theory, Enriched 2-categories, Categorification at roots of unity, Hopfological algebra
• Public URL: https://nottingham-repository.worktribe.com/output/3600094
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S0001870819305523