On $\lambda$-homomorphic skew braces

TL;DR

This paper investigates a special class of skew left braces called $\lambda$-homomorphic skew braces, providing necessary and sufficient conditions for their construction, and characterizing their structure, especially when the underlying group is free abelian of rank two.

Contribution

It introduces the concept of $\lambda$-homomorphic skew braces, establishes conditions for their existence, and characterizes their structure, particularly for free and free abelian groups.

Findings

Characterization of $\lambda$-homomorphic skew braces on free groups.

Construction methods for $\lambda$-homomorphic skew braces on free abelian groups.

Complete classification of such braces when the underlying abelian group has rank two.

Abstract

For a skew left brace $(G, \cdot, \circ)$, the map $\lambda : (G, \circ) \to \mathrm{Aut} \;(G, \cdot),~~a \mapsto \lambda_a$, where $\lambda_a(b) = a^{-1} \cdot (a \circ b)$ for all $a, b \in G$, is a group homomorphism. Then $\lambda$ can also be viewed as a map from $(G, \cdot)$ to $\mathrm{Aut}\; (G, \cdot)$, which, in general, may not be a homomorphism. We study skew left braces $(G, \cdot, \circ)$ for which $\lambda : (G, \cdot) \to \mathrm{Aut}\; (G, \cdot)$ is a homomorphism. Such skew left braces will be called $\lambda$-homomorphic. We formulate necessary and sufficient conditions under which a given homomorphism $\lambda : (G, \cdot) \to \mathrm{Aut}\; (G, \cdot)$ gives rise to a skew left brace, which, indeed, is $\lambda$-homomorphic. As an application, we construct skew left braces when $(G, \cdot)$ is either a free group or a free abelian group. We prove that any $\lambda$-homomorphic skew left brace is an extension of a trivial skew brace by a trivial skew brace. Special emphasis is given on $\lambda$-homomorphic skew left brace for which the image of $\lambda$ is cyclic. A complete characterization of such skew left braces on the free abelian group of rank two is obtained.