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Split Extensions and Actions of Bialgebras and Hopf Algebras

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.

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Acknowledgements

The author would like to warmly thank her supervisors Marino Gran and Joost Vercruysse for all the advice in the realization of this paper. Many thanks also to George Janelidze for the suggestion to explore split extensions and actions beyond the vector spaces case, in the context of monoidal categories. This led to a real improvement of this paper. The author also would like to thank Manuela Sobral for the invitation to the University of Coimbra and for the useful discussions. The author thanks the anonymous referee for his/her useful remarks and suggestions. The author’s research is supported by a FRIA (Fonds pour la formation à la recherche dans l’industrie et dans l’agriculture) doctoral grant no. 27485 of the Communauté française de Belgique.

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Communicated by George Janelidze.

Appendix

Appendix

This appendix contains five figures given below. The monoidal product is denoted by juxtaposition. Figure 13 is used in the proof of Proposition 5.16. By combining the diagrams of Figs. 14 and 15, we show that the structure of \(X \rtimes B\) as defined in (3.6) gives a bialgebra structure. Thanks to the commutativity of the diagram of Fig. 16, we can conclude that whenever X and B are associative bialgebras and (4.1) and (4.2) are satisfied \(m_{X\rtimes B}\) as defined in (3.6) is associative, which is a part of the proof of Lemma 4.3. Finally, the commutativity of Fig. 17 allows one to prove Proposition 3.9.

Fig. 13
figure13

Commutation of the diagram (A)

Fig. 14
figure14

The semi-direct product is a bialgebra: part 1

Fig. 15
figure15

The semi-direct product is a bialgebra: part 2

Fig. 16
figure16

The semi-direct product is associative

Fig. 17
figure17

Combination of the three diagrams (A), (B) and (C)

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Sterck, F. Split Extensions and Actions of Bialgebras and Hopf Algebras. Appl Categor Struct (2021). https://doi.org/10.1007/s10485-021-09659-5

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Keywords

  • (Non-associative) bialgebras
  • (Non-associative) Hopf algebras
  • Actions
  • Split extensions
  • Split short five lemma

Mathematics Subject Classification

  • 16T10
  • 16T05
  • 18C40
  • 18E99
  • 18M05
  • 17D99
  • 16S40