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Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
- Publisher
- Springer Journals
- Copyright
- Copyright © The Author(s), under exclusive licence to Springer Nature B.V. 2021
- ISSN
- 0927-2852
- eISSN
- 1572-9095
- DOI
- 10.1007/s10485-021-09659-5
- Publisher site
- See Article on Publisher Site
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.
Journal
Applied Categorical Structures – Springer Journals
Published: Apr 1, 2022
Keywords: (Non-associative) bialgebras; (Non-associative) Hopf algebras; Actions; Split extensions; Split short five lemma; 16T10; 16T05; 18C40; 18E99; 18M05; 17D99; 16S40