Department Mathematik
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Self-dual modules of semisimple Hopf algebras

Yevgenia Kashina Yorck Sommerhäuser
Yongchang Zhu

Abstract

We prove that, over an algebraically closed field of characteristic zero, a semisimple Hopf algebra that has a nontrivial self-dual simple module must have even dimension. This generalizes a classical result of W. Burnside. As an application, we show under the same assumptions that a semisimple Hopf algebra that has a simple module of even dimension must itself have even dimension.

Contents

  1. The Drinfel'd double
  2. The evaluation form
  3. The Frobenius-Schur theorem
  4. Self-dual modules
  5. Simple modules of even dimension
  6. The case of positive characteristic