Fusion and braiding in finite and affine Temperley-Lieb categories

Understand

Finite Temperley-Lieb (TL) algebras are diagram-algebra quotients of (the group algebra of) the famous Artin's braid group $B_N$, while the affine TL algebras arise as diagram algebras from a generalized version of the braid group.

• We study asymptotic `$N\to\infty$' representation theory of these quotients (parametrized by $q\in\mathbb{C}^{\times}$) from a perspective of braided monoidal categories.
• Using certain idempotent subalgebras in the finite and affine algebras, we construct infinite `arc' towers of the diagram algebras and the corresponding direct system of representation categories, with terms labeled by $N\in\mathbb{N}$.
• The corresponding direct-limit category is our main object of studies.