2020/04/12 by Valeriy G. Bardakov, Bardakov, Valeriy G., М. В. Нещадим +3
Decision Sciences · Mathematics · #16T25 #81R50 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Fuzzy and Soft Set Theory #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2004.05555
openalex publication_date 2020/04/12 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
For a skew left brace (G, ⋅, ∘), the map λ: (G, ∘) → Aut (G, ⋅),~~a ↦ λa, where λa(b) = a-1 ⋅ (a ∘ b) for all a, b ∈ G, is a group homomorphism. Then λ can also be viewed as a map from (G, ⋅) to Aut (G, ⋅), which, in general, may not be a homomorphism. We study skew left braces (G, ⋅, ∘) for which λ: (G, ⋅) → Aut (G, ⋅) 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, ⋅) → Aut (G, ⋅) gives rise to a skew left brace, which, indeed, is λ-homomorphic. As an application, we construct skew left braces when (G, ⋅) 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.