Tag - Temperley-Lieb algebras

The Drinfeld double of the Taft algebra, D_n, whose ground field contains nth roots of unity, has a known list of 2-dimensional irreducible modules. For each of such module V, we show that there is a well-defined action of the Temperley-Lieb algebra TL_k on the k-fold tensor product of V, and this action commutes with that of D_n. When V is self-dual and when k ≤ 2(n−1), we further establish a isomorphism between the centralizer algebra of D_n on V^⊗k, and TL_k. Our inductive argument uses a rank function on the TL diagrams, which is compatible with the nesting function introduced by Russell-Tymoczko.