Fusion and braiding in finite and affine Temperley-Lieb categories
by Azat Gainutdinov
Abstract: Finite and affine Temperley-Lieb algebras are ``diagram algebra'' quotients of an extended version of the Artin's braid group of type A_N. These quotients are also parametrised by non-zero complex number q. We study asymptotic representation theory (at N going to infinity) of these algebras from a perspective of braided monoidal categories: using certain idempotent subalgebras, we construct infinite ``arc'' towers of the diagram algebras and the corresponding direct system of representation categories, with terms labeled by N. The corresponding direct-limit category is our main object of studies.
For the case of the finite TL algebras, we prove that the direct-limit category is abelian and highest-weight at any non-zero q and endowed with braided monoidal structure. The most interesting result is when q is a root of unity where the representation theory is non-semisimple: the resulting braided monoidal categories are new and interestingly they are not rigid. We observe then a fundamental relation of these categories to a certain representation category of the Virasoro algebra and give a conjecture on the existence of a braided monoidal equivalence between the categories.
We also introduce a novel class of embeddings for the affine TL algebras and related new concept of fusion or bilinear N-graded tensor product of modules for these algebras. We prove that the fusion rules are stable with the index N of the tower and prove that the corresponding direct-limit category is endowed with an associative tensor product.