On faithfulness of the lifting for Hopf algebras and fusion categories

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 $p$.