## SciPost Submission Page

# Fusion and monodromy in the Temperley-Lieb category

### by Jonathan Belletête, Yvan Saint-Aubin

#### - Published as SciPost Phys. 5, 041 (2018)

### 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 |

Discipline: | Physics |

Subject area: | Mathematical Physics |

Approach: | Theoretical |

### 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

See full Ontology or Topics database.Published as SciPost Phys. 5, 041 (2018)

### Submission & Refereeing History

## Reports on this Submission

### Anonymous Report 1 on 2018-10-16 Invited Report

### Strengths

Same as in the previous report.

### Weaknesses

Same as in the previous report.

### Report

The authors have incorporated all the requested changes.

I recommend that this paper be accepted.

### Requested changes

None.