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arXiv · 2105.00428

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

Valeriy G. Bardakov·Vsevolod Gubarev
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Abstract

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. Recently, L. Guo, H. Lang, Y. Sheng [arXiv:2009.03492] 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.

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Valeriy G. Bardakov, Vsevolod Gubarev. 2022-01-22. Rota-Baxter groups, skew left braces, and the Yang-Baxter equation. https://doi.org/10.1016/j.jalgebra.2021.12.036

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