2016/06/14 by Gainutdinov, A. M., Saleur, H. · 1 citation
#Category Theory (math.CT) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1606.04530
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.