On twisting of finite-dimensional Hopf algebras

Eli Aljadeff, Pavel Etingof*, Shlomo Gelaki, Dmitri Nikshych

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

24 Scopus citations

Abstract

In this paper we study the properties of Drinfeld's twisting for finite-dimensional Hopf algebras. We determine how the integral of the dual to a unimodular Hopf algebra H changes under twisting of H. We show that the classes of cosemisimple unimodular, cosemisimple involutive, cosemisimple quasitriangular finite-dimensional Hopf algebras are stable under twisting. We also prove the cosemisimplicity of a coalgebra obtained by twisting of a cosemisimple unimodular Hopf algebra by two different twists on two sides (such twists are closely related to bi-Galois extensions), and describe the representation theory of its dual. Next, we define the notion of a non-degenerate twist for a Hopf algebra H, and set up a bijection between such twists for H and H*. This bijection is based on Miyashita-Ulbrich actions of Hopf algebras on simple algebras. It generalizes to the non-commutative case the procedure of inverting a non-degenerate skew-symmetric bilinear form on a vector space. Finally, we apply these results to classification of twists in group algebras and of cosemisimple triangular finite-dimensional Hopf algebras in positive characteristic, generalizing the previously known classification in characteristic zero.

Original languageEnglish
Pages (from-to)484-501
Number of pages18
JournalJournal of Algebra
Volume256
Issue number2
DOIs
StatePublished - 15 Oct 2002
Externally publishedYes

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