论文标题
融合2类中的刚性和可分离代数
Rigid and Separable Algebras in Fusion 2-Categories
论文作者
论文摘要
在整个量子代数和低维拓扑结构中，刚性单类别是无处不在的。我们研究了该概念的概括，即在任意单体2类中的刚性代数。刚性代数的示例包括$ g $的融合1类，以及$ g $ - 串联的融合1类。我们通过为存在右和左伴随的标准提供标准，从而探索了刚性代数的2类模块和双模模的属性。然后，我们考虑可分离的代数，这些代数特别表现良好。具体而言，考虑到融合2类别，我们证明了可分离代数上的模块和双模模的2类是有限的半神经。最后，我们在融合2类中定义了连接的刚性代数的尺寸，并证明这种代数是可以分开的，并且仅当其尺寸为非零时。
Rigid monoidal 1-categories are ubiquitous throughout quantum algebra and low-dimensional topology. We study a generalization of this notion, namely rigid algebras in an arbitrary monoidal 2-category. Examples of rigid algebras include $G$-graded fusion 1-categories, and $G$-crossed fusion 1-categories. We explore the properties of the 2-categories of modules and of bimodules over a rigid algebra, by giving a criterion for the existence of right and left adjoints. Then, we consider separable algebras, which are particularly well-behaved rigid algebras. Specifically, given a fusion 2-category, we prove that the 2-categories of modules and of bimodules over a separable algebra are finite semisimple. Finally, we define the dimension of a connected rigid algebra in a fusion 2-category, and prove that such an algebra is separable if and only if its dimension is non-zero.
