Fusion and monodromy in the Temperley-Lieb category

Graham and Lehrer (1998) introduced a Temperley-Lieb category $\widetilde{\mathsf{T}\mathsf{L}}$ whose objects are the non-negative integers and the morphisms in $\mathsf{H}\mathsf{o}\mathsf{m}(n,m)$ are the link diagrams from $n$ to $m$ nodes. The Temperley-Lieb algebra ${\mathsf{T}\mathsf{L}}_n$ is identified with $\mathsf{H}\mathsf{o}\mathsf{m}(n,n)$. The category $\widetilde{\mathsf{T}\mathsf{L}}$ is shown to be monoidal. We show that it is also a braided category by constructing explicitly a commutor. A twist is also defined on $\widetilde{\mathsf{T}\mathsf{L}}$. We introduce a module category ${\text{ Mod}}_{\widetilde{\mathsf{T}\mathsf{L}}}$ whose objects are functors from $\widetilde{\mathsf{T}\mathsf{L}}$ to ${\mathsf{V}\mathsf{e}\mathsf{c}\mathsf{t}}_{\mathbb{C}}$ and define on it a fusion bifunctor extending the one introduced by Read and Saleur (2007). We use the natural morphisms constructed for $\widetilde{\mathsf{T}\mathsf{L}}$ to induce the structure of a ribbon category on ${\text{ Mod}_{$\widetilde{\mathsf{T}\mathsf{L}}$}}(\beta=-q-q^{-1})$,when$q$ is not a root of unity. We discuss how the braiding on $\widetilde{\mathsf{T}\mathsf{L}}$ and integrability of statistical models are related. The extension of these structures to the family of dilute Temperley-Lieb algebras is also discussed.