Involutive Yang-Baxter groups

By Ferran Cedó, Eric Jespers, Á. Río. 2008

108 citations recorded by the source corpus.

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 of I-type. A group G of I-type is a group isomorphic to a subgroup of Fa n ⋊ Sym n so that the projection onto the first component is a bijective map, where Fa n is the free abelian group of rank n and Sym n is the symmetric group of degree n. The projection of G onto the second component Sym n we 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 group G, classify the groups of I-type with G as 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 of I-type (and hence infinitely many involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation) with a prescribed associated IYB group.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5b3d75eaec33a163cd2da04b967099103f8232e2

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Citations in the corpus

Braces and the Yang–Baxter Equation (2012, background): “A finite group G is said to be an involutive Yang–Baxter group (IYB group) if it is isomorphic to G(X, r) for some solution (X, r) of the YBE with X finite (see [4]).”
On skew braces (with an appendix by N. Byott and L. Vendramin) (2017, background): “In [44], Rump introduced braces, a new algebraic structure that turns out to be equivalent to bijective 1-cocycles; see [13, 27, 47].”
On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation (2015, background): “The same holds when (R,+, ·) is a left brace, and this solution is called the solution associated to left brace R, and will be denoted as (R, r) (for a reference see [18, 21, 44]).”
On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation (2017, methodology): “Thus, there are infinitely many solutions sharing the same G [CJdR10].”
Iterated matched products of finite braces and simplicity; new solutions of the Yang-Baxter equation (2016, background): “This class of solutions has received a lot of attention in recent years, see for example [8, 9, 12, 13, 14, 17, 18, 19, 20, 21, 25, 26, 29, 31, 39].”