## 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

### Citations:

Zbl 1371.16037; Zbl 1468.17026

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### References:

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