On actions of Drinfel'd doubles on finite dimensional algebras
Let $q$ be an $n^{th}$ root of unity for $n > 2$ and let $T_n(q)$ be the Taft (Hopf) algebra of dimension $n^2$. In 2001, Susan Montgomery and Hans-Jürgen Schneider classified all non-trivial $T_n(q)$-module algebra structures on an $n$-dimensional associative algebra $A$. They further showed tha...
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| Main Author | |
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| Format | Journal Article |
| Language | English |
| Published |
25.05.2018
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| Subjects | |
| Online Access | Get full text |
| DOI | 10.48550/arxiv.1805.10340 |
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| Summary: | Let $q$ be an $n^{th}$ root of unity for $n > 2$ and let $T_n(q)$ be the Taft
(Hopf) algebra of dimension $n^2$. In 2001, Susan Montgomery and Hans-Jürgen
Schneider classified all non-trivial $T_n(q)$-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 $T_n(q)$. 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 $u_q(\mathfrak{sl}_2)$. |
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| DOI: | 10.48550/arxiv.1805.10340 |