Fusion in the periodic Temperley-Lieb algebra: general definition of a bifunctor

The periodic Temperley-Lieb category consists of connectivity diagrams drawn on a ring with $N$ and $N^{\mathrm{\prime }}$ 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 $\mathsf{M}(N^{\mathrm{\prime }})\to \mathsf{M}(N)$ of the diagrams. Examples of modules that can be organised into families are those arising in the RSOS model and in the XXZ spin-$\frac{1}{2}$ 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.