Frobenius–Schur indicators and exponents of spherical categories

Abstract

We obtain two formulae for the higher Frobenius-Schur indicators: one for a spherical fusion category in terms of the twist of its center and the other one for a modular tensor category in terms of its twist. The first one is a categorical generalization of an analogous result by Kashina, Sommerhäuser, and Zhu for Hopf algebras, and the second one extends Bantay's 2nd indicator formula for a conformal field theory to higher degree. These formulae imply the sequence of higher indicators of an object in these categories is periodic. We define the notion of Frobenius-Schur (FS-)exponent of a pivotal category to be the global period of all these sequences of higher indicators, and we prove that the FS-exponent of a spherical fusion category is equal to the order of the twist of its center. Consequently, the FS-exponent of a spherical fusion category is a multiple of its exponent, in the sense of Etingof, by a factor not greater than 2. As applications of these results, we prove that the exponent and the dimension of a semisimple quasi-Hopf algebra H have the same prime divisors, which answers two questions of Etingof and Gelaki affirmatively for quasi-Hopf algebras. Moreover, we prove that the FS-exponent of H divides dim(H) 4 . In addition, if H is a group-theoretic quasi-Hopf algebra, the FS-exponent of H divides dim(H) 2 , and this upper bound is shown to be tight.

On the gauge equivalence of twisted quantum doubles of elementary abelian and extra-special 2-groups

“…We will see that there are at least 8, and no more than 20, gauge equivalence classes. This makes use of (2) to identify equivalence classes of quasi-bialgebras, together with Frobenius-Schur indicators and their higher analogs [NS3] to distinguish between equivalence classes. Now there is a canonical braiding of these quasiHopf algebras, and using the invariance of the canonical ribbon structure under braided tensor equivalences, which we establish separately, we show that the 20 gauge equivalence classes constitute a complete list of gauge equivalence classes of the quasi-triangular quasi-bialgebras under consideration.…”

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

“…The following equation (Eq. 11) is due to Ng and Schauenburg (see the proof of Theorem 5.5 of [18]). We present the proof for the sake of completeness.…”

“…where θ is the canonical twist of Z(C) and dim(C) = V∈Irr(C) |V| 2 (see [18,Theorem 4.1]). This formula plays an important role in Section 3.…”

Indicators of Tambara–Yamagami categories and Gauss sums

“…The Frobenius-Schur indicators of the simple objects of C can be used to define the Frobenius-Schur exponent of C, denoted FSexp(C). When C is the representation category of a quasi-Hopf algebra, FSexp(C) is equal to exp(C) or 2 exp(C) ([NS2], theorem 6.2) where exp(C) denotes the exponent of C in the sense of Etingof et.al. (see [E] and its references).…”