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Matched pairs and Yetter–Drinfeld braces

Part of: Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  03 April 2025

Davide Ferri Open the ORCID record for Davide Ferri [Opens in a new window]  and
Andrea Sciandra Open the ORCID record for Andrea Sciandra [Opens in a new window]
Davide Ferri
Affiliation:
Department of Mathematics “G. Peano”, University of Turin, via Carlo Alberto 10, Torino 10123, Italy Department of Mathematics & Data Science, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium e-mail: d.ferri@unito.it Davide.Ferri@vub.be
Andrea Sciandra*
Affiliation:
Department of Mathematics “G. Peano”, University of Turin, via Carlo Alberto 10, Torino 10123, Italy
*
e-mail: andrea.sciandra@unito.it
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Abstract

It is proven that a matched pair of actions on a Hopf algebra H is equivalent to the datum of a Yetter–Drinfeld brace, which is a novel structure generalizing Hopf braces. This [-30pt] improves a theorem by Angiono, Galindo, and Vendramin, originally stated for cocommutative Hopf braces. These Yetter–Drinfeld braces produce Hopf algebras in the category of Yetter–Drinfeld modules over H, through an operation that generalizes Majid’s transmutation. A characterization of Yetter–Drinfeld braces via 1-cocycles, in analogy to the one for Hopf braces, is given.

Every coquasitriangular Hopf algebra H will be seen to yield a Yetter–Drinfeld brace, where the additional structure on H is given by the transmutation. We compute explicit examples of Yetter–Drinfeld braces on the Sweedler’s Hopf algebra, on the algebras $E(n)$, on $\mathrm {SL}_{q}(2)$, and an example in the class of Suzuki algebras.

Keywords

Yetter–Drinfeld bracesmatched pairsHopf bracesquasitriangular Hopf algebrasbraiding operators

MSC classification

Primary: 16T05: Hopf algebras and their applications
Secondary: 16T25: Yang-Baxter equations
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 37
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The authors were partially supported by the Ministry for University and Research (MUR) within the National Research Project (PRIN 2022) “Structures for Quivers, Algebras and Representations” (SQUARE) Project 2022S97PMY. The first-named author was partially funded by the Vrije Universiteit Brussel Bench Fee for a Joint Doctoral Project, grant number OZR4257, and partially supported through the FWO Senior Research Project G004124N.

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