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

· pdf
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

· pdf
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 Report on Fusion and monodromy in the TemperleyLieb category