S 4 symmetry of 6j symbols and Frobenius–Schur indicators in rigid monoidal C* categories

Abstract

We show that a left-rigid monoidal C * -category with irreducible monoidal unit is also a sovereign and spherical category. Defining a Frobenius--Schur type indicator we obtain selection rules for the fusion coefficients of irreducible objects. As a main result we prove S 4 -invariance of 6j-symbols in such a category.

Higher Frobenius-Schur indicators for pivotal categories

“…Thus E (2) V is (at most) a scalar, which coincides with its trace; that computing the trace in a different way leads to the indicator formulas from [13] was shown in [17]. An endomorphism of C(V ∨ , V ) conjugate to E (2) V is also used to describe the degree two indicator in [5] (and to define E (2) V in [17]). The maps E (n) V have been studied in connection with 3-manifold invariants by Gelfand and Kazhdan [7].…”

“…In the last case, proving that ν 2 (V ) ∈ {0, ±1} uses the result of Etingof, Nikshych, and Ostrik [4] that the module category of a semisimple complex quasi-Hopf algebra is a pivotal monoidal category. A different proof based on this pivotal structure and the description of indicators in [5] was given in [17].…”

“…The classical (degree two) Frobenius-Schur indicator ν 2 (V ) of an irreducible representation V of a finite group G has been generalized to Frobenius-Schur indicators of simple modules of semisimple Hopf algebras by Linchenko and Montgomery [12], to certain C * -fusion categories by Fuchs, Ganchev, Szlachányi, and Vescernyés [5], and further to simple objects in pivotal (or sovereign) categories by Fuchs and Schweigert [6]; Mason and Ng [13] treated the case of simple modules over semisimple quasi-Hopf algebras. As in the classical case, the indicator always takes one of the values 0, ±1, and is related to the question if and how the representation in consideration is self-dual.…”

Central invariants and higher indicators for semisimple quasi-Hopf algebras

“…The (degree 2) Frobenius-Schur indicator ν 2 (V ) of an irreducible representation V of a finite group G has been generalized to simple modules of semisimple Hopf algebras by Linchenko and Montgomery [LM00], to certain C * -fusion categories by Fuchs, Ganchev, Szlachányi, and Vecsernyés [FGSV99], and to simple modules of semisimple quasi-Hopf algebras by Mason and the first author [MN05]. A more general version of the Frobenius-Schur Theorem holds for the simple modules of semisimple Hopf algebras or even quasi-Hopf algebras.…”

The half-twist forUq(g) representations

“…The following definition of the Frobenius-Schur indicator for a pivotal category was given by [Ng and Schauenburg 2007] (see also [Linchenko and Montgomery 2000] for the case of Hopf algebras and [Fuchs et al 1999] for C * sovereign categories).…”