Classifying coalgebra split extensions of Hopf algebras

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 $p\ge 3$ an infinite family of non-isomorphic Hopf algebras of dimension $4p$ is constructed. The groups of automorphisms of these Hopf algebras are also described.

21 pages; substantial changes from previous version; to appear in J. Algebra Appl

References in corpus (2)

• Cocycle deformations for Hopf algebras with a coalgebra projection
• Crossed product of Hopf algebras