The Classif ication of All Crossed Products H 4 # 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] we classify all coalgebra split extensions of H4 by k[Cn], where Cn is the cyclic group of order n and H4 is Sweedler’s 4-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras H4#k[Cn] by explicitly computing two classifying objects: the cohomological ‘group’ H(k[Cn], H4) and Crp(k[Cn], H4) := the set of types of isomorphisms of all crossed products H4#k[Cn]. More precisely, all crossed products H4#k[Cn] are described by generators and relations and classified: they are 4n-dimensional quantum groups H4n,λ,t, parameterized by the set of all pairs (λ, t) consisting of an arbitrary unitary map t : Cn → C2 and an n-th root λ of ±1. As an application, the group of Hopf algebra automorphisms of H4n,λ,t is explicitly described.