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The Classification of All Crossed Products $H_4 \# k[C_{n}]$
Symmetry, integrability and geometry: methods and applications, Tome 10 (2014) Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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Ana-Loredana Agore; Costel-Gabriel Bontea; Gigel Militaru. The Classification of All Crossed Products $H_4 \# k[C_{n}]$. Symmetry, integrability and geometry: methods and applications, Tome 10 (2014). http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a48/

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