Split extensions and actions of bialgebras and Hopf algebras

TL;DR

This paper develops a generalized framework for split extensions and actions in non-associative bialgebras and Hopf algebras, establishing their equivalence and proving a key lemma.

Contribution

It introduces a new notion of split extension for non-associative bialgebras and Hopf algebras, showing their equivalence to actions and extending classical results.

Findings

Split extensions are equivalent to actions in non-associative bialgebras.

The concept extends to non-associative Hopf algebras with similar equivalence.

The Split Short Five Lemma holds for these split extensions.

Abstract

We introduce a notion of split extension of (non-associative) bialgebras which generalizes the notion of split extension of magmas introduced by M. Gran, G. Janelidze and M. Sobral. We show that this definition is equivalent to the notion of action of (non-associative) bialgebras. We particularize this equivalence to (non-associative) Hopf algebras by defining split extensions of (non-associative) Hopf algebras and proving that they are equivalent to actions of (non-associative) Hopf algebras. Moreover, we prove the validity of the Split Short Five Lemma for these kinds of split extensions, and we examine some examples.