On $λ$-homomorphic skew braces

For a skew left brace $(G, \cdot, \circ)$, the map $λ: (G, \circ) \to \mathrm{Aut} \;(G, \cdot),~~a \mapsto λ_a$, where $λ_a(b) = a^{-1} \cdot (a \circ b)$ for all $a, b \in G$, is a group homomorphism. Then $λ$ 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 $λ: (G, \cdot) \to \mathrm{Aut}\; (G, \cdot)$ is a homomorphism. Such skew left braces will be called $λ$-homomorphic. We formulate necessary and sufficient conditions under which a given homomorphism $λ: (G, \cdot) \to \mathrm{Aut}\; (G, \cdot)$ gives rise to a skew left brace, which, indeed, is $λ$-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 $λ$-homomorphic skew left brace is an extension of a trivial skew brace by a trivial skew brace. Special emphasis is given on $λ$-homomorphic skew left brace for which the image of $λ$ is cyclic. A complete characterization of such skew left braces on the free abelian group of rank two is obtained.

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