Computing the Frobenius-schur Indicator for Abelian Extensions of Hopf Algebras

Let H be a finite-dimensional semisimple Hopf algebra. Recently it was shown in [LM] that a version of the Frobenius-Schur theorem holds for Hopf algebras, and thus that the Schur indicator ν(χ) of the character χ of a simple H-module is well-defined; this fact for the special case of Kac algebras was shown in [FGSV]. In this paper we show that for an important class of non-trivial Hopf algebras, ν(χ) is a computable invariant. The Hopf algebras we consider are all abelian extensions; as a special case, they include the Drinfeld double of a group algebra. In addition to finding a general formula for the indicator, we also study when it is always positive. In particular we prove that the indicator is always positive for the Drinfeld double of the symmetric group, generalizing the classical result for the symmetric group itself. As a first step in proving this, we show that the indicator can be computed by means of a “local indicator”. Finally we show that work of the first author on the classification of Hopf algebras of dimension 16 can be somewhat shortened using indicators rather than K0. It is likely that the indicator will be useful in other problems on the classification of semisimple Hopf algebras. Moreover, Schur indicators play a role in various aspects of conformal field theory; see work of Bantay [B1] [B2]. We first introduce some notation. Throughout, H will be a finite-dimensional Hopf algebra over an algebraically closed field k of characteristic not 2, with comultiplication ∆ : H −→ H ⊗H , via h 7→ ∑