[Paper Review] A note on set-theoretic solutions of the Yang-Baxter equation

Agata Smoktunowicz|arXiv (Cornell University)|Dec 21, 2015

Advanced Topics in Algebra32 references3 citations

TL;DR

This paper establishes that every finite non-degenerate involutive set-theoretic solution of the Yang-Baxter equation with a cube-free permutation group cardinality is a multipermutation solution. It further proves that finite left braces with odd cardinality and the identity (−a)·b = −(a·b) are two-sided braces, hence Jacobson radical rings, and shows that semidirect and wreath products of finite multipermutation-level braces remain of finite multipermutation level.

ABSTRACT

This paper shows that every finite non-degenerate involutive set theoretic solution (X,r) of the Yang-Baxter equation whose symmetric group has cardinality which a cube-free number is a multipermutation solution. Some properties of finite braces are also investigated (Theorems 3, 5 and 11). It is also shown that if A is a left brace whose cardinality is an odd number and (-a) b=-(ab) for all a, b A, then A is a two-sided brace and hence a Jacobson radical ring. It is also observed that the semidirect product and the wreath product of braces of a finite multipermutation level is a brace of a finite multipermutation level.

Motivation & Objective

• To characterize finite non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation with cube-free permutation group cardinality.
• To investigate structural properties of finite left braces, particularly conditions under which they become two-sided braces.
• To determine whether semidirect and wreath products of finite multipermutation-level braces preserve the finite multipermutation property.
• To explore connections between braces, Jacobson radical rings, and the Yang-Baxter equation via group-theoretic constructions.

Proposed method

• Uses the correspondence between left braces and non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation.
• Applies group-theoretic techniques to analyze the permutation group 𝒢(X,r) of a solution (X,r), focusing on its cardinality being cube-free.
• Employs the radical chain construction A^{(n)} and A^n for left braces to study multipermutation levels.
• Utilizes the semidirect product construction N ⋊ H for left braces, with H acting on N via automorphisms, to analyze multipermutation levels.
• Applies the wreath product construction G ≀ H as a semidirect product W ⋊ H, where W consists of finitely supported functions from H to G.
• Leverages known results on brace ideals and adjoint groups to prove closure properties under algebraic operations.

Experimental results

Research questions

• RQ1 Under what conditions is a finite non-degenerate involutive set-theoretic solution of the Yang-Baxter equation a multipermutation solution?
• RQ2 When does a finite left brace with odd cardinality and the identity (−a)·b = −(a·b) become a two-sided brace?
• RQ3 Does the semidirect product of two finite multipermutation-level left braces result in a brace of finite multipermutation level?
• RQ4 Is the wreath product of two finite multipermutation-level left braces also of finite multipermutation level?
• RQ5 Can every finite solvable group be embedded into the adjoint group of a finite multipermutation-level left brace?

Key findings

• Every finite non-degenerate involutive set-theoretic solution (X,r) with |𝒢(X,r)| cube-free is a multipermutation solution.
• If A is a finite left brace with odd cardinality and satisfies (−a)·b = −(a·b) for all a,b ∈ A, then A is a two-sided brace and hence a Jacobson radical ring.
• The semidirect product A ⋊ B of two finite multipermutation-level left braces A and B is itself a brace of finite multipermutation level.
• The wreath product A ≀ B of two finite multipermutation-level left braces A and B is also a brace of finite multipermutation level.
• If a semidirect product N ⋊ H has finite multipermutation level, then both N and H must individually have finite multipermutation level.
• The wreath product G ≀ H has finite multipermutation level if and only if both G and H have finite multipermutation level.