模范畴的Grothendieck环：局部化、分层与根
Grothendieck Rings of Module Categories: Localizations, Strata, and Radicals
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中文总结 AI 辅助
本文研究非退化辫子融合范畴的模范畴的Grothendieck环，通过定义Drinfeld中心中étale代数的卷积和全中心的同构，建立了环的上三角分层结构，并利用转移矩阵和Gram矩阵计算了多个例子中的层环与根，揭示了非交换性和非零根的存在。
中文摘要 AI 辅助
对于一个非退化的辫子融合范畴 $\mathcal B$，我们研究 $\mathcal B$-模范畴的Grothendieck $\mathbb Z_+$-环。我们在Drinfeld中心 $\mathcal Z(\mathcal B)$ 中定义了étale代数的卷积，并证明了模范畴的相对张量积的全中心与其全中心的卷积典范同构。模Grothendieck环具有由 $\mathcal B$ 的局部化偏序集给出的典范上三角分次。我们通过双陪集端点块和群环值转移矩阵描述每个局部化层，得到了显式的夹层乘法公式。当 $\mathcal B$ 是一个融合范畴的中心时，最大层的转移矩阵是Lagrangian代数的Gram矩阵，因此它们的类之间的线性关系产生根。我们在经典有限几何、有限群的Drinfeld双重、例外WZW范畴和Haagerup中心的例子中计算了层环及其复化的根。这些例子展示了具有半单复化的非交换模Grothendieck环，以及在中间层和最大复化层中均存在非零根。
英文摘要
For a non-degenerate braided fusion category $\mathcal B$, we study the Grothendieck $\mathbb Z_+$-ring of $\mathcal B$-module categories. We define a convolution of étale algebras in the Drinfeld center $\mathcal Z(\mathcal B)$ and show that the full center of a relative tensor product of module categories is canonically isomorphic to the convolution of their full centers. The module Grothendieck ring has a canonical upper-triangular grading by the poset of localizations of $\mathcal B$. We describe each localization stratum by double-coset endpoint blocks and a group-ring-valued transition matrix, obtaining an explicit sandwich multiplication formula. When $\mathcal B$ is the center of a fusion category, the transition matrix of the maximal stratum is the Gram matrix of the Lagrangian algebras, so linear relations among their classes give rise to radicals. We compute stratum rings and the radicals of their complexifications in examples from classical finite geometry, Drinfeld doubles of finite groups, exceptional WZW categories, and the Haagerup center. The examples exhibit noncommutative module Grothendieck rings with semisimple complexifications, as well as nonzero radicals in both intermediate and maximal complexified strata.
