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On structure groups of set-theoretic solutions to the Yang-Baxter equation. (English) Zbl 1423.16034

Authors’ abstract: This paper explores the structure groups \(G_{(X,r)}\) of finite non-degenerate set-theoretic solutions \((X,r)\) to the Yang-Baxter equation. Namely, we construct a finite quotient \(\overline {G}_{(X,r)}\) of \(G_{(X,r)}\), generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: if \(X\) injects into \(G_{(X,r)}\), then it also injects into \(\overline {G}_{(X,r)}\). We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization of \(G_{(X,r)}\). We show that multipermutation solutions are the only involutive solutions with diffuse structure groups; that only free abelian structure groups are bi-orderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: bi-orderable, left-orderable, abelian, free abelian and torsion free.

MSC:

16T25 Yang-Baxter equations
20N02 Sets with a single binary operation (groupoids)
06F15 Ordered groups
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