On faithfulness of the lifting for Hopf algebras and fusion categories

Etingof, Pavel

doi:10.2140/ant.2018.12.551

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On faithfulness of the lifting for Hopf algebras and fusion categories

Pavel Etingof

Vol. 12 (2018), No. 3, 551–569

DOI: 10.2140/ant.2018.12.551

Abstract

We use a version of Haboush’s theorem over complete local Noetherian rings to prove faithfulness of the lifting for semisimple cosemisimple Hopf algebras and separable (braided, symmetric) fusion categories from characteristic $p$ to characteristic zero, showing that, moreover, any isomorphism between such structures can be reduced modulo $p$ . This fills a gap in our earlier work. We also show that lifting of semisimple cosemisimple Hopf algebras is a fully faithful functor, and prove that lifting induces an isomorphism on Picard and Brauer–Picard groups. Finally, we show that a subcategory or quotient category of a separable multifusion category is separable (resolving an open question from our earlier work), and use this to show that certain classes of tensor functors between lifts of separable categories to characteristic zero can be reduced modulo $p$ .

To Alexander Kirillov, Jr. on his 50th birthday with admiration

Keywords

lifting, Hopf algebra, tensor category, separable

Mathematical Subject Classification 2010

Primary: 16T05

Milestones

Received: 27 April 2017

Revised: 18 October 2017

Accepted: 18 December 2017

Published: 12 June 2018

Authors

Pavel Etingof
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States

Vol. 12, No. 3, 2018