Fusion and monodromy in the Temperley-Lieb category

Abstract

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

32 pages, 4 figures