Frobenius-Schur Indicators and Exponents of Spherical Categories
| dc.creator | Ng, Siu-Hung | |
| dc.creator | Schauenburg, Peter | |
| dc.date | 2005-12-31 | |
| dc.date | 2006-08-05 | |
| dc.date.accessioned | 2026-07-07T07:50:59Z | |
| dc.date.available | 2026-07-07T07:50:59Z | |
| dc.description | 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, Sommerhauser, 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. | |
| dc.description | 32p. LaTex file with macros and figures. Some typos and Thm 8.4 in v2 have been corrected. The current Thm 8.4 is a combined result of Thms 8.4 and 8.5 in version 2 | |
| dc.identifier | https://arxiv.org/abs/math/0601012 | |
| dc.identifier | http://arxiv.org/abs/math/0601012 | |
| dc.identifier | Adv. Math. 211 (2007) no. 1, 34--71 | |
| dc.identifier | doi:10.1016/j.aim.2006.07.017 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/125359 | |
| dc.subject | Quantum Algebra | |
| dc.subject | Category Theory | |
| dc.title | Frobenius-Schur Indicators and Exponents of Spherical Categories | |
| dc.type | text |