模范畴、整性与埃及分数
Modular categories, integrality and Egyptian fractions
摘要：
Etingof，Nikshych和Ostrik的一个著名结果是：任意固定秩$n$都存在1000个不等的整模范畴.这是从埃及分数表示中的最大分母1的双指数界得出的。一个简单的计算机搜索方法来分类秩$n$整模范畴使用此界快速地消除计算机的内存（对于$n\geq 7$）。我们使用一种改进的策略：找到模范畴蕴含完整性的一般条件，并在这些有限的条件下研究分类问题。第一个条件是扭矩阵的阶为2，3，4或6，我们得到了这几类模范畴的一个相当完整的刻画.第二个条件是单位对象是唯一的简单非自对偶对象，这等价于奇维。在这种情况下，我们得到一个（线性）改进的界限，并采用数论技术，以获得一个分类的秩最多为11的奇数维模块化类别。
英文摘要：
It is a well-known result of Etingof, Nikshych and Ostrik that there are finitely many inequivalent integral modular categories of any fixed rank $n$. This follows from a double-exponential bound on the maximal denominator in an Egyptian fraction representation of 1. A naïve computer search approach to the classification of rank $n$ integral modular categories using this bound quickly overwhelms the computer's memory (for $n\geq 7$). We use a modified strategy: find general conditions on modular categories that imply integrality and study the classification problem in these limited settings. The first such condition is that the order of the twist matrix is 2,3,4 or 6 and we obtain a fairly complete description of these classes of modular categories. The second condition is that the unit object is the only simple non-self-dual object, which is equivalent to odd-dimensionality. In this case we obtain a (linear) improvement on the bounds and employ number-theoretic techniques to obtain a classification for rank at most 11 for odd-dimensional modular categories.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
