Aspects of the category SKB of skew braces

Abstract

We examine the pointed protomodular category SKB of left skew braces. We study the notion of commutator of ideals in a left skew brace. Notice that in the literature, “product” of ideals of skew braces is often considered. We show that the so-called (Huq = Smith) condition holds for left skew braces. Finally, we give a set of generators for the commutator of two ideals, and prove that every ideal of a left skew brace has a centralizer.

KEYWORD:

• Brace
• commutator
• (Smith = Huq) condition
• Mal’tsev category
• protomodular category
• skew brace
• Yang-Baxter equation

2020MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16T25
• 18E13
• 20N99

Additional information

Funding

The second author is partially supported by Ministero dell’Istruzione, dell’Università e della Ricerca (Progetto di ricerca di rilevante interesse nazionale “Categories, Algebras: Ring-Theoretical and Homological Approaches (CARTHA)”), Fondazione Cariverona (Research project “Reducing complexity in algebra, logic, combinatorics - REDCOM” within the framework of the programme Ricerca Scientifica di Eccellenza 2018), and the Department of Mathematics “Tullio Levi-Civita” of the University of Padua (Research programme DOR1828909 “Anelli e categorie di moduli”).