Survey on matched pairs of groups-an elementary approach to the ESS-LYZ theory

By M. Takeuchi. 2003

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Matched pairs approach to set theoretic solutions of the Yang–Baxter equation (2005, methodology): “We recall that a braided group, [26], is a pair (G, σ), where G is a group and σ : G × G → G × G is a map such that the left and the right actions induced by σ make (G, G) a matched pair of groups with condition M3.”
On skew braces (with an appendix by N. Byott and L. Vendramin) (2017, background): “11 is implicit in the work of Lu, Yan and Zhu, see [38, Theorem 2] and [53].”
Post-groups, (Lie-)Butcher groups and the Yang–Baxter equation (2023, background): “[55] A matched pair of groups is a triple (G,H, σ), where G and H are groups and σ : G × H → H ×G, (a, h) 7→ (a ⇀ h, a ↼ h) is a map satisfying the following conditions: eG ⇀ h = h, (36) a ⇀ (b ⇀ h) = (ab) ⇀ h, (37) (ab) ↼ h = ( a ↼ (b ⇀ h) ) (b ↼ h), (38) a ↼ eH = a, (39) (a ↼ h) ↼ k = a ↼ (hk), (4”
Iterated matched products of finite braces and simplicity; new solutions of the Yang-Baxter equation (2016, background): “Note that matched products of groups also appear in the context of solutions of the YBE, see for example the survey of Takeuchi [38] and [22].”
Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity (2010, background): “There are numerous works on set-theoretic solutions and related structures, of which a relevant selection for the interested reader is [39, 23, 34, 12, 27, 18, 17, 21, 24, 35, 36, 20, 21, 22, 8].”