Central invariants and higher indicators for semisimple quasi-Hopf algebras

By Siu-Hung Ng, Peter Schauenburg. 2005

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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, Sommerhauser, 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/78653a5f1fbc4cd26a980ab20765231602d3f952

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Citations in the corpus

HOPF MONOIDS IN THE CATEGORY OF SPECIES (2012, background): “In the recent Hopf algebra literature, the operations h 7→ h are called Hopf powers [46] or Sweedler powers [36, 37].”
Higher Frobenius-Schur indicators for pivotal categories (2005, methodology): “In another paper [ 14 ] we will study the pivotal fusion categories of modules over a semisimple complex quasi-Hopf algebra.”
On the Gauge Equivalence of Twisted Quantum Doubles of Elementary Abelian and Extra-Special 2-Groups (2006, methodology): “ 24 Appendix 2Let H= Dω(G) where G= E 8,D 8 or Q 8; χan irreducible representation of Hand ν(n) χ the nth Frobenius-Schur indicator of χ. The indicators can be computed from the formulas of [MN2] and [NS1]. The table below displays the ﬁrst 15 Frobenius-Schur indicators for each of the irreducible r”