Hecke algebras for set-theoretical solutions to the Yang–Baxter equation

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We define a concept of Hecke algebra for structure groups of set-theoretical solutions to the Yang–Baxter equation. As a comparison to Artin–Tits groups of spherical type, we study some properties of this construction, while also highlighting some differences that appear, which shows a difference between finite Coxeter groups and the “Coxeter-like” group introduced by Dehornoy. We also relate this definition to known constructions on solutions (retractions). Finally, we study a particular case related to Torus Knot groups and Complex Reflexion groups.