Involutive Yang-Baxter groups

Abstract

In 1992 Drinfeld posed the question of finding the set-theoretic solutions of the Yang-Baxter equation. Recently, Gateva-Ivanova and Van den Bergh and Etingof, Schedler and Soloviev have shown a group-theoretical interpretation of involutive non-degenerate solutions. Namely, there is a one-to-one correspondence between involutive non-degenerate solutions on finite sets and groups ofII-type. A groupG\mathcal {G}ofII-type is a group isomorphic to a subgroup ofFan⋊Symn\mathrm {Fa}_n\rtimes \mathrm {Sym}_nso that the projection onto the first component is a bijective map, whereFan\mathrm {Fa}_nis the free abelian group of ranknnandSymn\mathrm {Sym}_{n}is the symmetric group of degreenn. The projection ofG\mathcal {G}onto the second componentSymn\mathrm {Sym}_nwe call an involutive Yang-Baxter group (IYB group). This suggests the following strategy to attack Drinfeld’s problem for involutive non-degenerate set-theoretic solutions. First classify the IYB groups and second, for a given IYB groupGG, classify the groups ofII-type withGGas associated IYB group. It is known that every IYB group is solvable. In this paper some results supporting the converse of this property are obtained. More precisely, we show that some classes of groups are IYB groups. We also give a non-obvious method to construct infinitely many groups ofII-type (and hence infinitely many involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation) with a prescribed associated IYB group.