The Classification of All Crossed Products H₄#k[Cn]

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2014
• Main Authors: Agore, A.L., Bontea, C.G., Militaru, G.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2014
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/146697
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: The Classification of All Crossed Products H₄#k[Cn] / A.L. Agore, C.G. Bontea, G. Militaru // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 14 назв. — англ.

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' H²(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.