Title: Fusion in the periodic Temperley-Lieb algebra: general definition of a bifunctor Show Chinese title

Title: 周期 Temperley-Lieb 代数中的融合：双函子的一般定义

Abstract: The periodic Temperley-Lieb category consists of connectivity diagrams drawn on a ring with $N$ and $N'$ nodes on the outer and inner boundary, respectively. We consider families of modules, namely sequences of modules $\mathsf{M}(N)$ over the enlarged periodic Temperley-Lieb algebra for varying values of $N$, endowed with an action $\mathsf{M}(N') \to \mathsf{M}(N)$ of the diagrams. Examples of modules that can be organised into families are those arising in the RSOS model and in the XXZ spin-$\frac12$ chain, as well as several others constructed from link states. We construct a fusion product which outputs a family of modules from any pair of families. Its definition is inspired from connectivity diagrams drawn on a disc with two holes. It is thus defined in a way to describe intermediate states in lattice correlation functions. We prove that this fusion product is a bifunctor, and that it is distributive, commutative, and associative. Show Chinese abstract

Abstract: 周期性Temperley-Lieb范畴由在环上绘制的连通图组成，环的外边界和内边界分别有$N$和$N'$个节点。 我们考虑模块族，即对于不同值的$N$，关于扩展的周期性Temperley-Lieb代数的模块序列$\mathsf{M}(N)$，并赋予图的行动$\mathsf{M}(N') \to \mathsf{M}(N)$。 可以组织成族的模块示例包括RSOS模型和XXZ自旋$\frac12$链中出现的模块，以及从连接态构造的其他几个模块。 我们构造了一个融合乘积，可以从任何一对族中输出一个模块族。 它的定义受到在有两个孔的圆盘上绘制的连通图的启发。 因此，它是以描述格子关联函数中的中间状态的方式定义的。 我们证明这个融合乘积是一个双函子，并且它是分配律的、交换的和结合的。