Basic Hopf algebras and symmetric bimodules

Katerina Hristova, Vanessa Miemietz

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Abstract

Motivated by the so-called H-cell reduction theorems, we investigate certain classes of bicategories which have only one H-cell apart from possibly the identity. We show that H_0-simple quasi fiab bicategories with unique H-cell H_0 are fusion categories. We further study two classes of non-semisimple quasi-fiab bicategories with a single H-cell apart from the identity. The first is $\cH_A$, indexed by a finite-dimensional radically graded basic Hopf algebra A, and the second is $\cG_A$, consisting of symmetric projective A-A-bimodules. We show that $\cH_A$ can be viewed as a 1-full subbicategory of $\cG_A$ and classify simple transitive birepresentations for $\cG_A$. We point out that the number of equivalence classes of the latter is finite, while that for $\cH_A$ is generally not.
Original languageEnglish
Article number107328
JournalJournal of Pure and Applied Algebra
Volume227
Issue number7
Early online date16 Jan 2023
DOIs
Publication statusPublished - Jul 2023

Keywords

  • Bicategory
  • Hopf algebra
  • Symmetric bimodule

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