Classifying coalgebra split extensions of Hopf algebras

TL;DR

This paper classifies all coalgebra split extensions of a Hopf algebra by the Sweedler's 4-dimensional Hopf algebra, explicitly describing the crossed products and their automorphisms, including new examples in characteristic p.

Contribution

It provides a complete classification of coalgebra split extensions of Hopf algebras by H_4, including explicit generators, relations, and automorphism groups, with new examples in characteristic p.

Findings

Classified all Hopf algebra extensions by H_4.

Explicitly described crossed products via generators and relations.

Constructed infinite families of non-isomorphic Hopf algebras in characteristic p.

Abstract

For a given Hopf algebra $A$ we classify all Hopf algebras $E$ that are coalgebra split extensions of $A$ by $H_4$, where $H_4$ is the Sweedler's 4-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras $A # H_4$ by computing explicitly two classifying objects: the cohomological 'group' ${\mathcal H}^{2} (H_4, A)$ and $\textsc{C}\textsc{r}\textsc{p} (H_4, A) :=$ the set of types of isomorphisms of all crossed products $A # H_4$. All crossed products $A #H_4$ are described by generators and relations and classified: they are parameterized by the set ${\mathcal Z}{\mathcal P} (A)$ of all central primitive elements of $A$. Several examples are worked out in detail: in particular, over a field of characteristic $p \geq 3$ an infinite family of non-isomorphic Hopf algebras of dimension $4p$ is constructed. The groups of automorphisms of these Hopf algebras are also described.