Simplicity of indecomposable set-theoretic solutions of the Yang–Baxter equation

Abstract

This paper aims to deepen the theory of bijective non-degenerate set-theoretic solutions of the Yang–Baxter equation, not necessarily involutive, by means of q-cycle sets. We entirely focus on the finite indecomposable ones, among which we especially study the class of simple solutions. In particular, we provide a group-theoretic characterization of these solutions, including their permutation groups. Finally, we deal with some open questions.

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

“…1) If X is an indecomposable cycle set with G(X) abelian and X = p α1 1 ...p αn n , where p 1 , ..., p n are distinct prime numbers, then f pl(X) = α 1 + ... + α n (see [8,Theorem 4.4] for more details). These cycle sets have been explicitly classified if mpl(X) = 2 (see [22]) and if G(X) is cyclic (see [23]).…”

“…2) Every indecomposable cycle set X having finite multipermutation level is a cycle set of finite primitive level (see [8,Corollary 4.5]) and mpl(X) ≤ f pl(X) (several concrete examples of indecomposable cycle sets belonging to the multipermutation ones are contained, for example, in [21] and [14]).…”

“…This implicitly suggested a new perspective to study indecomposable solutions, focusing on the imprimitive blocks systems. Actually, the family of indecomposable involutive solutions of finite primitive level is an unexplored topic, except for a recent paper of the author [9], where a generalization on the non-involutive case is also given. The first main result of the paper provides a group-theoretic characterization of indecomposable solutions of finite primitive level by means of a standard subgroup of the associated permutation group called the displacements group, already considered in [21] to study multipermutation solutions and in [5] to study latin solutions.…”

Indecomposable solutions of the Yang-Baxter equation with permutation group of sizes $pq$ and $p^2q$

“…Classifying indecomposable solutions seems to be a more approachable objective, there are, for example, only 36 indecomposable solutions of size 10. This class of solutions has been intensively studied with many recent results [10,9,20,24,23,11,8,18,21,7] In this paper we give a classification of all indecomposable solutions whose permutation group (see Definition 2.2) has size pq, or is abelian or dihedral and of size p 2 q, where p and q are distinct primes.…”

On derived-indecomposable solutions of the Yang--Baxter equation

“…Many authors focused on studying and classifying such solutions with a special emphasis on those which are also involutive and non-degenerate, i.e. solutions (X, r) such that r 2 = id, and if we write r(x, y) = (σ x (y), τ y (x)) then the maps σ x and τ y are bijective, for all x, y ∈ X; see [6,7,8,9,18,21,25,33,34,36,37,39]. On the other hand, almost nothing is known about indecomposable solutions which are non-degenerate and non-involutive (see [13,26,32]).…”