Affine Temperley-Lieb algebra at roots of unity and logarithmic conformal field theory

Temperley-Lieb algebras of finite/affine type are rank-one quotients of the finite/affine Hecke algebras and known in the theory of invariants as Schur-Weyl duals to the Lusztig's specialization of the (affine) quantum algebra for sl(2). On the physics side, the TL algebras appear in a transfer-matrix formulation of statistical lattice models, like XXZ spin-chains. At q roots of unity, the lattice models enjoy an interesting property that taking the continuum limit gives a logarithmic conformal field theory, which is a non-semisimple representation space for the Virasoro algebra. Taking these limits in a rigorous way is usually poorly understood. In the talk, I will present an explicit inductive system construction for finite TL algebras at a root of unity that gives the Virasoro in the limit. On the abstract grounds, I will also discuss my recent result on a connection between representation theory of affine (and finite) TL algebras at all roots of unity cases and the Virasoro algebra at critical values of central charge.