整模数据与同余
Integral modular data and congruences
摘要：
我们计算所有的融合代数的对称有理$S$ -矩阵到12维。其中只有两个可以用作模数据中的$S$ -矩阵：$\mathbb{Z}/2\mathbb{Z}$和$S_3$的量子双的$S$ -矩阵。几乎所有这些都满足一定的一致性，这有一些有趣的含义，例如他们的程度。我们还明确地给出了一个无穷序列的模块数据与合理的$S$ -和$T$ -矩阵，既不是张量积较小的模块数据，也不是有限群的量子双打的$S$ -矩阵。对于有限群序列（$S_3,D_4,Q_8,S_4$的某些次直积），我们证明了它们的量子偶的$S$ -矩阵的合理性.
英文摘要：
We compute all fusion algebras with symmetric rational $S$-matrix up to dimension 12. Only two of them may be used as $S$-matrices in a modular datum: the $S$-matrices of the quantum doubles of $\mathbb{Z}/2\mathbb{Z}$ and $S_3$. Almost all of them satisfy a certain congruence which has some interesting implications, for example for their degrees. We also give explicitly an infinite sequence of modular data with rational $S$- and $T$-matrices which are neither tensor products of smaller modular data nor $S$-matrices of quantum doubles of finite groups. For some sequences of finite groups (certain subdirect products of $S_3,D_4,Q_8,S_4$), we prove the rationality of the $S$-matrices of their quantum doubles.
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