Central invariants and higher indicators for semisimple quasi-Hopf algebras

Abstract

Abstract. In this paper, we define the higher Frobenius-Schur (FS-)indicators for finite-dimensional modules of a semisimple quasi-Hopf algebra H via the categorical counterpart developed in a 2005 preprint. When H is an ordinary Hopf algebra, we show that our definition coincides with that introduced by Kashina, Sommerhäuser, and Zhu. We find a sequence of gauge invariant central elements of H such that the higher FS-indicators of a module V are obtained by applying its character to these elements. As an application, we show that FS-indicators are sufficient to distinguish the four gauge equivalence classes of semisimple quasi-Hopf algebras of dimension eight corresponding to the four fusion categories with certain fusion rules classified by Tambara and Yamagami. Three of these categories correspond to well-known Hopf algebras, and we explicitly construct a quasi-Hopf algebra corresponding to the fourth one using the Kac algebra. We also derive explicit formulae for FS-indicators for some quasi-Hopf algebras associated to group cocycles.

Higher Frobenius-Schur indicators for pivotal categories

“…In the present paper we define and study higher Frobenius-Schur indicators ν n (V ) for an object V of a k-linear pivotal monoidal category C. We do this with a view towards the categories of modules over semisimple complex quasi-Hopf algebras, though we will only give the (quite involved) explicit formulas and examples for that case in another paper [14].…”

“…Finally, in Section 7, we show that the nth Frobenius-Schur indicator of an object V in a semisimple k-linear pivotal monoidal category isthe pivotal trace of a natural endomorphism FS (n) V , called the Frobenius-Schur endomorphism. In another paper [14] we will study the pivotal fusion categories of modules over a semisimple complex quasi-Hopf algebra. In this case the Frobenius-Schur endomorphisms correspond to central gauge invariants in the quasi-Hopf algebras.…”

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

“…8 is equal to the right-hand side of Eq. 1 if V ∈ Rep(H) for some semisimple Hopf algebra H (see [20,Remark 4.3]). They also generalized the "third formula" of Kashina et al [8,Section 6.4, Corollary] to objects of spherical fusion category: If C is a spherical fusion category, we have…”

Examples of non-$FSZ$ $p$-groups for primes greater than three

“…The study of the representation categories of semisimple Hopf algebras, and many other more general contexts, have brought forth an interesting invariant of monoidal categories known as (higher) Frobenius-Schur indicators [1,2,3,5,9,12,13,14,15,16,17,19]. These form generalizations of the classical Frobenius-Schur indicators for a finite group G, which for a character χ of G over C and any m ∈ N are defined by When applied to the Hopf algebra D(G), the Drinfel'd double of the finite group G over C, these indicators can be expressed entirely in group theoretical terms.…”