TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.5.4.041
TI  - Fusion and monodromy in the Temperley-Lieb category
PY  - 2018/10/31
UR  - https://www.scipost.org/SciPostPhys.5.4.041
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 5
IS  - 4
SP  - 041
A1  - Belletête, Jonathan
AU  - Saint-Aubin, Yvan
AB  - 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.
ER  -

@Article{10.21468/SciPostPhys.5.4.041,
	title={{Fusion and monodromy in the Temperley-Lieb category}},
	author={Jonathan Belletête and Yvan Saint-Aubin},
	journal={SciPost Phys.},
	volume={5},
	pages={041},
	year={2018},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.5.4.041},
	url={https://scipost.org/10.21468/SciPostPhys.5.4.041},
}