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The Classification of All Crossed Products $H_4 \# k[C_{n}]$
Ana-Loredana Agoreab,
Costel-Gabriel Bonteaac,
Gigel Militaruc a Faculty of Engineering, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium
b Department of Applied Mathematics, Bucharest University of Economic Studies, Piata Romana 6, RO-010374 Bucharest 1, Romania
c Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, RO-010014 Bucharest 1, Romania
Abstract:
Using the computational approach introduced in [Agore A.L., Bontea C.G., Militaru G.,
J. Algebra Appl. 12 (2013), 1250227, 24 pages] we classify all coalgebra split extensions of
$H_4$ by
$k[C_n]$, where
$C_n$ is the cyclic group of order
$n$ and
$H_4$ is Sweedler's
$4$-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras
$H_4 \# k[C_{n}]$ by explicitly computing two classifying objects: the cohomological ‘group’
${\mathcal H}^{2} ( k[C_{n}], H_4)$ and
$\mathrm{Crp} ( k[C_{n}], H_4):=$ the set of types of isomorphisms of all crossed products
$H_4 \# k[C_{n}]$. More precisely, all crossed products
$H_4 \# k[C_n]$ are described by generators and relations and classified: they are
$4n$-dimensional quantum groups
$H_{4n, \lambda, t}$, parameterized by the set of all pairs
$(\lambda, t)$ consisting of an arbitrary unitary map
$t : C_n \to C_2$ and an
$n$-th root
$\lambda$ of
$\pm 1$. As an application, the group of Hopf algebra automorphisms of
$H_{4n, \lambda, t}$ is explicitly described.
Keywords:
crossed product of Hopf algebras; split extension of Hopf algebras.
MSC: 16T10;
16T05;
16S40 Received: November 18, 2013; in final form
April 18, 2014; Published online
April 23, 2014
Language: English
DOI:
10.3842/SIGMA.2014.049