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Fusion and monodromy in the Temperley-Lieb category

by Jonathan Belletête, Yvan Saint-Aubin

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Submission summary

Authors (as registered SciPost users): Yvan Saint-Aubin
Submission information
Preprint Link:  (pdf)
Date accepted: 2018-10-23
Date submitted: 2018-10-04 02:00
Submitted by: Saint-Aubin, Yvan
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


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.

Published as SciPost Phys. 5, 041 (2018)

Reports on this Submission

Anonymous Report 1 on 2018-10-16 (Invited Report)


Same as in the previous report.


Same as in the previous report.


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

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