Hopf braces and semi-abelian categories

TL;DR

This paper studies cocommutative Hopf braces, showing they form a semi-abelian category, and explores their algebraic properties, homological lemmas, and subcategory structures, linking them to solutions of the quantum Yang-Baxter equation.

Contribution

It establishes that cocommutative Hopf braces form a semi-abelian and strongly protomodular category, extending classical homological results to this new algebraic structure.

Findings

Category of cocommutative Hopf braces is semi-abelian.

Abelian objects are commutative and cocommutative Hopf algebras.

Skew braces form a Birkhoff subcategory and a localization.

Abstract

Hopf braces have been introduced as a Hopf-theoretic generalization of skew braces. Under the assumption of cocommutativity, these algebraic structures are equivalent to matched pairs of actions on Hopf algebras, that can be used to produce solutions of the quantum Yang-Baxter equation. We prove that the category of cocommutative Hopf braces is semi-abelian and strongly protomodular. In particular, this implies that the main homological lemmas known for groups, Lie algebras and other classical algebraic structures also hold for cocommutative Hopf braces. Abelian objects are commutative and cocommutative Hopf algebras, that form an abelian Birkhoff subcategory of the category of cocommutative Hopf braces. Moreover, we show that the full subcategories of "primitive Hopf braces" and of "skew braces" form an hereditary torsion theory in the category of cocommutative Hopf braces, and that "skew braces" are also a Birkhoff subcategory and a localization of the latter category. Finally, we describe central extensions and commutators for cocommutative Hopf braces.