Central invariants and higher indicators for semisimple quasi-Hopf algebras
Central invariants and higher indicators for semisimple quasi-Hopf algebras
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DOI：
10.1090/s0002-9947-07-04276-6
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发表时间：
2005-08
影响因子：
1.3
通讯作者：
S. Ng;P. Schauenburg
中科院分区：
文献类型：
--
作者：
S. Ng;P. Schauenburg
本文通过2005年预印本中发展的范畴对应定义了半单拟Hopf代数H的有限维模的高阶Frobenius-Schur(FS-)指标。当H是普通的Hopf代数时，我们证明了我们的定义与Kashina，Sommerhauser和朱引入的定义一致。我们找到了H的一个规范不变中心元序列，使得模V的高阶FS-指标是通过将它的性质应用于这些元而得到的。作为应用，我们证明了FS-指标足以区分与Tambara和Yamagami分类的具有一定融合规则的四个融合范畴对应的八维半单拟Hopf代数的四个规范等价类。其中三个范畴对应于著名的Hopf代数，我们利用Kac代数显式地构造了对应于第四个范畴的拟Hopf代数。我们还给出了一些与群余循环相关的拟Hopf代数的FS-指标的显式公式。
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.