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