Hopf formulae for homology of skew braces

TL;DR

This paper develops new algebraic formulas for computing the homology of skew braces using non-abelian homological algebra, providing explicit sequences and characterizations of central extensions.

Contribution

It introduces Hopf formulae for skew brace homology and characterizes central extensions algebraically, advancing the understanding of skew brace structure.

Findings

Derived Hopf formulae for skew brace homology

Established explicit Stallings-Stammbach exact sequence

Characterized central extensions algebraically

Abstract

The variety of skew braces contains several interesting subcategories as subvarieties, as for instance the varieties of radical rings, of groups and of abelian groups. In this article the methods of non-abelian homological algebra are applied to establish some new Hopf formulae for homology of skew braces, where the coefficient functors are the reflectors from the variety of skew braces to each of the three above-mentioned subvarieties. The corresponding central extensions of skew braces are characterized in purely algebraic terms, leading to some new results, such as an explicit Stallings-Stammbach exact sequence associated with any exact sequence of skew braces, and a new result concerning central series.