Let $\mathcal{C}(\mathfrak{g},k)$ be the unitary modular tensor categories arising from the representation theory of quantum groups at roots of unity for arbitrary simple finite-dimensional complex Lie algebra $\mathfrak{g}$ and positive integer levels $k$. Here we classify nondegenerate fusion subcategories of the modular tensor categories of local modules $\mathcal{C}(\mathfrak{g},k)_R^0$ where $R$ is the regular algebra of Tannakian $\text{Rep}(H)\subset\mathcal{C}(\mathfrak{g},k)_\text{pt}$. For $\mathfrak{g}=\mathfrak{so}_5$ we describe the decomposition of $\mathcal{C}(\mathfrak{g},k)_R^0$ into prime factors explicitly and as an application we classify relations in the Witt group of nondegenerately braided fusion categories generated by the equivalency classes of $\mathcal{C}(\mathfrak{so}_5,k)$ and $\mathcal{C}(\mathfrak{g}_2,k)$ for $k\in\mathbb{Z}_{\geq1}$.

中文翻译：

---

D 类局部模块的模张量类别的素数分解

令 $\mathcal{C}(\mathfrak{g},k)$ 是由任意简单有限维复数李代数 $\mathfrak{g} 的单位根上的量子群的表示理论产生的幺正模张量范畴$ 和正整数级别 $k$。在这里，我们对局部模块 $\mathcal{C}(\mathfrak{g},k)_R^0$ 的模张量类别的非退化融合子类别进行分类，其中 $R$ 是 Tannakian $\text{Rep}( H)\subset\mathcal{C}(\mathfrak{g},k)_\text{pt}$。对于 $\mathfrak{g}=\mathfrak{so}_5$，我们明确地描述了 $\mathcal{C}(\mathfrak{g},k)_R^0$ 分解为质因子的过程，并且作为一个应用，我们对关系进行分类在由 $\mathcal{C}(\mathfrak{so}_5,k)$ 和 $\mathcal{C}(\mathfrak{g}_2,