TY - JOUR

T1 - On λ-homomorphic skew braces

AU - Bardakov, Valeriy G.

AU - Neshchadim, Mikhail V.

AU - Yadav, Manoj K.

N1 - Funding Information: The authors thank the referee for reading the manuscript with utmost care and providing some very useful suggestions and ideas. The first author is supported by the Ministry of Science and Higher Education of Russia (agreement No. 075-02-2021-1392 ). The first and second named authors acknowledge the support of RFBR (project No. 19-01-00569 ). The third named author acknowledges the support of DST-RSF Grant INT/RUS/RSF/P-2 . Publisher Copyright: © 2021 Elsevier B.V.

PY - 2022/6

Y1 - 2022/6

N2 - 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 a lot of skew left braces (of infinite order) on free groups and free abelian groups. 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. We also obtain set-theoretic solutions of the Yang-Baxter equation corresponding to the skew braces we construct in this paper.

AB - 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 a lot of skew left braces (of infinite order) on free groups and free abelian groups. 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. We also obtain set-theoretic solutions of the Yang-Baxter equation corresponding to the skew braces we construct in this paper.

KW - Left brace

KW - Skew left brace

KW - Symmetric skew brace

KW - Yang-Baxter equation

KW - λ-Cyclic

KW - λ-Homomorphic

UR - http://www.scopus.com/inward/record.url?scp=85120156949&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2021.106961

DO - 10.1016/j.jpaa.2021.106961

M3 - Article

AN - SCOPUS:85120156949

VL - 226

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 6

M1 - 106961

ER -