Fusion and monodromy in the Temperley-Lieb category.

Graham and Lehrer (1998) introduced a Temperley-Lieb category (TL) over tilde whose objects are the non-negative integers and the morphisms in Hom(n, m) are the link diagrams from n to m nodes. The Temperley-Lieb algebra TLn is identified with Hom (n, n). The category (TL) over tilde 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 (TL) over tilde. We introduce a module category Mod((TL) over tilde) whose objects are functors from (TL) over tilde to Vect(C) and define on it a fusion bifunctor extending the one introduced by Read and Saleur (2007). We use the natural morphisms constructed for (TL) over tilde to induce the structure of a ribbon category on Mod((TL) over tilde)(beta = -q-q(-1)), when q is not a root of unity. We discuss how the braiding on (TL) over tilde and integrability of statistical models are related. The extension of these structures to the family of dilute Temperley-Lieb algebras is also discussed. Copyright J. Belletete and Y. Saint-Aubin. This work is licensed under the Creative Commons Attribution 4.0 International License. Published by the SciPost Foundation.