Rota—Baxter groups, skew left braces, and the Yang—Baxter equation

By Valeriy G. Bardakov, Vsevolod Gubarev. Journal of Algebra, 2021

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Braces were introduced by W. Rump in 2006 as an algebraic system related to the quantum Yang—Baxter equation. In 2017, L. Guarnieri and L. Vendramin defined for the same purposes a more general notion of a skew left brace. In 2021, L. Guo, H. Lang and Y. Sheng gave a definition of what is a Rota—Baxter operator on a group. We connect these two notions as follows. It is shown that every Rota— Baxter group gives rise to a skew left brace. Moreover, every skew left brace can be injectively embedded into a Rota—Baxter group. When the additive group of a skew left brace is complete, then this brace is induced by a Rota—Baxter group. We interpret some notions of the theory of skew left braces in terms of Rota—Baxter operators.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a888be01d45c82eb9ae86822f8b85f1d20d635c4