A family of irretractable square-free solutions of the Yang–Baxter equation

Abstract

A new family of non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation is constructed. All these solutions are strong twisted unions of multipermutation solutions of multipermutation level at most two. A large subfamily consists of irretractable and square-free solutions. This subfamily includes a recent example of Vendramin [38, Example 3.9], who first gave a counterexample to Gateva-Ivanova's Strong Conjecture [19, Strong Conjecture 2.28(I)]. All the solutions in this subfamily are new counterexamples to Gateva-Ivanova's Strong Conjecture and also they answer a question of Cameron and Gateva-Ivanova [21, Open Questions 6.13 (II)(4)]. It is proved that the natural left brace structure on the permutation group of the solutions in this family has trivial socle. Properties of the permutation group and of the structure group associated to these solutions are also investigated. In particular, it is proved that the structure groups of finite solutions in this subfamily are not poly-(infinite cyclic) groups.

On the indecomposable involutive set-theoretic solutions of the Yang-Baxter equation of prime-power size

“…It is known that every finite cycle set is non-degenerate (see [21,Theorem 2]). Moreover, the image σ(X) of the map σ : X −→ Sym(X), x → σ x can be endowed with an induced binary operation σ x · σ y := σ x·y which satisfies (1). The third author [21] showed that (σ(X), ·) is a (nondegenerate) cycle set if and only if (X, ·) is non-degenerate.…”

About a question of Gateva-Ivanova and Cameron on square-free set-theoretic solutions of the Yang-Baxter equation

“…2) The unique square-free left cycle set of size 3 and level 2 is given by σ 1 = σ 2 := id X and σ 3 := (1 2) and the group of automorphism is generated by σ 3 . Since σ [1] (1) = σ [1] (2), necessarilyN 2 > 3. Now, let X := {1, 2, 3, 4} be the left cycle set given by σ 1 = σ 2 := (3 4) σ 3 = σ 4 := (1 2).…”

“…The existence of this bijective correspondence allows to move the study of involutive non-degenerate solutions to non-degenerate left cycle sets. In this context, we prove the inequality (1) by a mixture of two well-known extensiontools of left cycle sets: the one-sided extension of left cycle sets, developed in terms of set-theoretic solutions [9] by Etingof, Schedler and Soloviev, and the dynamical extension of left cycle sets developed in [19] by Vendramin. In the last section we will see that the same approach is useful to construct further interesting examples of left cycle sets: referring to [20,Problem 19], we provide several counterexamples to the Gateva-Ivanova's Conjecture, in addition to those obtained in [19,1,3].…”

The Matched Product of the Solutions to the Yang–Baxter Equation of Finite Order

“…A solution r on X is said to be involutive if r 2 = id. Such solutions have been intensively studied, see [20,12,35,2,5] just to name a few. In particular, Rump in [29] introduced braces, ring-like structures, for studying involutive non-degenerate solutions.…”