3 years ago

# On actions of Drinfel'd doubles on finite dimensional algebras

Zachary Cline

Publication date: Available online 12 November 2018

Source: Journal of Pure and Applied Algebra

Author(s): Zachary Cline

##### 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 $uq(sl2)$.

DOI: S0022404918302901

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