WEAKLY GROUP-THEORETICAL AND SOLVABLE FUSION CATEGORIES
WEAKLY GROUP-THEORETICAL AND SOLVABLE FUSION CATEGORIES
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DOI：
10.1016/j.aim.2010.06.009
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发表时间：
2008-09
影响因子：
1.7
通讯作者：
P. Etingof;D. Nikshych;V. Ostrik
中科院分区：
文献类型：
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作者：
P. Etingof;D. Nikshych;V. Ostrik
我们引入了两类新的融合范畴，它们是通过一定的过程从有限群中获得的——弱群论范畴和可解范畴。这些是融合类别，森田相当于任意可解有限群的迭代扩展（在融合类别的世界中）。弱群论范畴具有整数维数，并且所有已知的整数维数融合范畴都是弱群论范畴。我们的主要结果是，弱群论范畴 C 具有强 Frobenius 性质（即，不可分解的 C 模范畴中任何简单对象的维数除以 C 的维数），并且任何维度最多有两个素因数的融合范畴都是可解的（有限群的伯恩赛德定理的分类模拟）。这对于融合类别的分类和给定维度的半简单 Hopf 代数具有强大的应用。特别是，我们证明任何整数维度 <84 的融合类别都是弱群论的（即来自有限群论），并给出维度 pqr 和 pq2 的半单 Hopf 代数的完整分类，其中 p、q、r 是不同的素数。
We introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups – weakly group-theoretical categories and solvable categories. These are fusion categories that are Morita equivalent to iterated extensions (in the world of fusion categories) of arbitrary, respectively solvable finite groups. Weakly group-theoretical categories have integer dimension, and all known fusion categories of integer dimension are weakly group-theoretical. Our main results are that a weakly group-theoretical category C has the strong Frobenius property (i.e., the dimension of any simple object in an indecomposable C-module category divides the dimension of C), and that any fusion category whose dimension has at most two prime divisors is solvable (a categorical analog of Burnside's theorem for finite groups). This has powerful applications to classification of fusion categories and semsisimple Hopf algebras of a given dimension. In particular, we show that any fusion category of integer dimension <84 is weakly group-theoretical (i.e. comes from finite group theory), and give a full classification of semisimple Hopf algebras of dimensions pqr and pq2, where p,q,r are distinct primes.