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Rota-Baxter groups, skew left braces, and the Yang-Baxter equation. (English) Zbl 1492.17019

In relation to the famous Yang-Baxter equation, there is the notion of skew braces (introduced by [L. Guarnieri and L. Vendramin, Math. Comput. 86, 2519–2534 (2017; Zbl 1371.16037)] as a generalization of braces) and the notion of Rota-Baxter groups (introduced by [L. Guo, H. Lang, and Y. Sheng, Adv. Math. 387, Article ID 107834, 34 p. (2021; Zbl 1468.17026)] as an analogue of Rota-Baxter operators defined on algebras).
A skew (left) brace is a triplet \((G,\cdot,\circ)\) for which \((G,\cdot)\) and \((G,\circ)\) are groups satisfying the relation \[ x\circ( y\cdot z) = (x\circ y)\cdot x^{-1} \cdot (x\circ z)\mbox{ for all }x,y,z\in G. \] A Rota-Baxter group is a group \((G,\cdot)\) endowed with a map \(B: G\rightarrow G\) (a so-called Rota-Baxter operator of weight \(1\)) such that \[ B(x)B(y) = B(xB(x)yB(x)^{-1})\mbox{ for all }x,y\in G. \] The paper under review studies the connection between skew braces and Rota-Baxter groups. The authors showed that every Rota-Baxter group \((G,\cdot,B)\) gives rise to a skew brace \((G,\cdot,\circ)\) by defining \[ x \circ y = xB(x)yB(x)^{-1}\mbox{ for all }x,y\in G \] (Proposition 3.1). Conversely, every skew brace \((G,\cdot,\circ)\) for which \((G,\cdot)\) is complete arises in this way (Proposition 3.12). In general, every skew brace may be embedded into a Rota-Baxter group (Theorem 3.6).
After proving the above connection between skew braces and Rota-Baxter groups, the authors discussed various notions from the theory of skew braces and interpret them via Rota-Baxter groups.

MSC:

17B38 Yang-Baxter equations and Rota-Baxter operators
16T25 Yang-Baxter equations

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References:

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