Hopf braces and semi-abelian categories
Marino Gran, Andrea Sciandra
TL;DR
The paper establishes that the category $\mathsf{HBR}_{\mathrm{coc}}$ of cocommutative Hopf braces is semi-abelian and strongly protomodular, providing a robust homological framework for these Hopf-theoretic structures. By leveraging the equivalence with matched pairs of actions and bijective 1-cocycles, it develops semi-direct product decompositions, regularity, and exactness results closely mirroring classical group-like categories. It identifies abelian objects as commutative cocommutative Hopf algebras (a Birkhoff subcategory) and constructs a hereditary torsion theory with primitive braces and skew braces, yielding a localization of the main category. The work further analyzes central extensions and Huq commutators, showing a Smith–Huq compatibility and enabling a detailed understanding of commutator theory in Hopf braces. These results connect Hopf braces with classical algebraic structures and provide a solid categorical foundation for their (co)homology and Yang–Baxter equation applications.
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.