The Classification of All Crossed Products $H_4\mathrm{\#}k[C_n]$

Using the computational approach introduced in [Agore A.L., Bontea C.G., Militaru G., J. Algebra Appl. 12 (2013), 1250227, 24 pages, arXiv:1207.0411] 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 $\text{CRP}(k[C_n],H_4):=$ the set of types of isomorphisms of all crossed products $H_4\mathrm{\#}k[C_n]$. More precisely, all crossed products $H_4\mathrm{\#}k[C_n]$ 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 $(\lambda, t)$consistingofanarbitraryunitarymap$t : C_n \to C_2$andan$n$-throot$\lambda$of$\pm 1$. As an application, the group of Hopf algebra automorphisms of $H_{4n,\lambda ,t}$ is explicitly described.