On actions of Drinfel'd doubles on finite dimensional algebras

Abstract

Let q be an nth root of unity for n>2 and let Tn(q) be the Taft (Hopf) algebra of dimension n2. In 2001, Susan Montgomery and Hans-Jürgen Schneider classified all non-trivial Tn(q)-module algebra structures on an n-dimensional associative algebra A. They further showed that each such module structure extends uniquely to make A a module algebra over the Drinfel'd double of Tn(q). We explore what it is about the Taft algebras that leads to this uniqueness, by examining actions of (the Drinfel'd double of) Hopf algebras H “close” to the Taft algebras on finite-dimensional algebras analogous to A above. Such Hopf algebras H include the Sweedler (Hopf) algebra of dimension 4, bosonizations of quantum linear spaces, and the Frobenius–Lusztig kernel uq(sl2).