SciPost Phys. 8, 019 (2020) ·
published 5 February 2020

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The boundary seam algebras $\mathsf{b}_{n,k}(\beta=q+q^{1})$ were introduced by MorinDuchesne, Ridout and Rasmussen to formulate algebraically a large class of boundary conditions for twodimensional statistical loop models. The representation theory of these algebras $\mathsf{b}_{n,k}(\beta=q+q^{1})$ is given: their irreducible, standard (cellular) and principal modules are constructed and their structure explicited in terms of their composition factors and of nonsplit short exact sequences. The dimensions of the irreducible modules and of the radicals of standard ones are also given. The methods proposed here might be applicable to a large family of algebras, for example to those introduced recently by Flores and Peltola, and Cramp\'e and Poulain d'Andecy.
SciPost Phys. 5, 041 (2018) ·
published 31 October 2018

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Graham and Lehrer (1998) introduced a TemperleyLieb category $\mathsf{\widetilde{TL}}$ whose objects are the nonnegative integers and the morphisms in $\mathsf{Hom}(n,m)$ are the link diagrams from $n$ to $m$ nodes. The TemperleyLieb 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 ${\text{ Mod}_{\mathsf{\widetilde{TL}}}}(\beta=qq^{1})$, 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 TemperleyLieb algebras is also discussed.
Prof. SaintAubin: "First, we would like to thanks..."
in Submissions  report on Fusion and monodromy in the TemperleyLieb category