The Classification of All Crossed Products $H_4 \# k[C_{n}]$

TL;DR

This paper classifies all crossed product Hopf algebras formed from Sweedler's $H_4$ and cyclic groups $C_n$, explicitly describing their structure, automorphisms, and parameterizations.

Contribution

It provides a complete classification of crossed products $H_4 owtie k[C_n]$ using cohomological methods and explicit generators and relations, introducing new quantum groups $H_{4n, \lambda, t}$.

Findings

Classification of all such crossed products as $4n$-dimensional quantum groups.

Explicit description of automorphism groups of these quantum groups.

Parameterization of these quantum groups by pairs $(\lambda, t)$.

Abstract

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 $H_4 \# k[C_{n}]$. 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.