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 N and N' N ′ nodes on the outer and inner boundary, respectively. We consider families of modules, namely sequences of modules \mathsf{M}(N) 𝖬 ( N ) over the enlarged periodic Temperley–Lieb algebra for varying values of N N , endowed with an action \mathsf{M}(N') \to \mathsf{M}(N) 𝖬 ( N ′ ) → 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 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.