Some Computations of Frobenius–Schur Indicators of the Regular Representations of Hopf Algebras

Abstract

We study Frobenius-Schur indicators of the regular representations of finite-dimensional semisimple Hopf algebras, especially group-theoretical ones. Those of various Hopf algebras are computed explicitly. In view of our computational results, we formulate the theorem of Frobenius for semisimple Hopf algebras and give some partial results on this problem.

Frobenius–Schur indicators in Tambara–Yamagami categories

“…In [20], the author showed that Frobenius theorem holds for H under this condition, and hence the result follows. We note that Theorem 4.4 and Corollary 4.5 cover cases where there does not exist such a Hopf algebra H .…”

“…For a pivotal fusion category C, we define ν n (C) by [20]. The results of Section 3 yield the following formula for ν n of Tambara-Yamagami categories.…”

“…One might ask whether ν n (C) has the same property. Motivated by this question, the author introduced the following definition in [20]. Definition 4.3.…”

“…In [20], the author showed that Frobenius theorem holds for various semisimple Hopf algebras and conjectured that it holds for every semisimple Hopf algebra. The author also pointed out that there exist many semisimple quasi-Hopf algebras for which Frobenius theorem does not hold.…”

“…For a pivotal fusion category C, the author [20] has introduced the symbol ν n (C) as a certain weighted sum of n-th Frobenius-Schur indicators of simple objects of C. In Section 4, we study ν n (C) for Tambara-Yamagami categories C. We also argue and give some partial results for C, which are motivated by the classical theorem of Frobenius.…”