Pointed Hopf algebras are a wide class of Hopf algebras, including group algebras and enveloping algebras of Lie algebras. In this talk, based on a recent work with Susan Montgomery, we study actions of pointed Hopf algebras on simple algebras. These actions are known to be inner, as in the case of Skolem-Noether theorem. We try to give explicit descriptions, whenever possible, and consider Taft algebras, their Drinfeld doubles and some quantum groups.
The Drinfeld double of the Taft algebra, Dn, 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 TLk on the k-fold tensor product of V, and this action commutes with that of Dn. When V is self-dual and when k ≤ 2(n−1), we further establish a isomorphism between the centralizer algebra of Dn on V⊗k, and TLk. Our inductive argument uses a rank function on the TL diagrams, which is compatible with the nesting function introduced by Russell-Tymoczko.
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