# Fusion and monodromy in the Temperley-Lieb category

### Submission summary

 As Contributors: Yvan Saint-Aubin Arxiv Link: https://arxiv.org/abs/1802.09203v3 Date accepted: 2018-10-23 Date submitted: 2018-10-04 Submitted by: Saint-Aubin, Yvan Submitted to: SciPost Physics Domain(s): Theoretical Subject area: Mathematical Physics

### Abstract

Graham and Lehrer (1998) introduced a Temperley-Lieb category $\mathsf{\widetilde{TL}}$ 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 $\mathsf{\widetilde{TL}}$ 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 $\mathsf{\widetilde{TL}}$. We introduce a module category ${\text{ Mod}_{\mathsf{\widetilde{TL}}}}$ whose objects are functors from $\mathsf{\widetilde{TL}}$ 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 $\mathsf{\widetilde{TL}}$ to induce the structure of a ribbon category on ${\text{ Mod}_{\mathsf{\widetilde{TL}}}}(\beta=-q-q^{-1})$, when $q$ is not a root of unity. We discuss how the braiding on $\mathsf{\widetilde{TL}}$ and integrability of statistical models are related. The extension of these structures to the family of dilute Temperley-Lieb algebras is also discussed.

### Ontology / Topics

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### Submission & Refereeing History

Resubmission 1802.09203v3 on 4 October 2018
Submission 1802.09203v2 on 3 August 2018

## Reports on this Submission

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### Report

The authors have incorporated all the requested changes.
I recommend that this paper be accepted.

### Requested changes

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