Finite Temperley-Lieb (TL) algebras are diagram-algebra quotients of (the
group algebra of) the famous Artin's braid group BN​, while the affine TL
algebras arise as diagram algebras from a generalized version of the braid
group. We study asymptotic `N→∞' representation theory of these
quotients (parametrized by q∈C×) 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∈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 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 we obtain at different roots of unity 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. This should have powerful applications to the study of
the `continuum' limit of critical statistical mechanics systems based on the TL
algebra.
We also introduce a novel class of embeddings for the affine Temperley-Lieb
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. We also
study the braiding properties of this affine TL fusion.Comment: 50p