[Paper Review] On the Yang-Baxter equation and left nilpotent left braces

Ferran Cedó, Tatiana Gateva-Ivanova+1 more|Edinburgh Research Explorer|Jan 26, 2016

Finite Group Theory Research6 references3 citations

TL;DR

This paper investigates the structure of finite non-degenerate involutive solutions to the Yang-Baxter equation via left braces and their associated structure groups. It proves that the structure group of a finite non-trivial solution cannot be an Engel group, providing a rich class of poly-ℤ groups that are not Engel, and establishes embeddings of finite solutions into finite braces and braided groups.

ABSTRACT

We study non-degenerate involutive set-theoretic solutions (X,r) of the Yang-Baxter equation, we call them simply solutions. We show that the structure group G(X,r) of a finite non-trivial solution (X,r) cannot be an Engel group. It is known that the structure group G(X,r) of a finite multipermutation solution (X,r) is a poly-Z group, thus our result gives a rich source of examples of braided groups and left braces G(X,r) which are poly-Z groups but not Engel groups. We also show that a finite solution of the Yang-Baxter equation can be embedded in a convenient way into a finite brace and into a finite braided group. For a left brace A, we explore the close relation between the multipermutation level of the solution associated with it and the radical chain $A^{(n+1)}=A^{(n)}* A$ introduced by Rump.

Motivation & Objective

• To understand the group-theoretic properties of structure groups associated with finite non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation.
• To investigate the conditions under which such structure groups can be Engel groups, particularly in the context of left braces.
• To establish that finite solutions can be embedded into finite braces and finite braided groups, enriching the class of realizable algebraic structures.
• To clarify the connection between the multipermutation level of a solution and the radical chain in left braces as defined by Rump.

Proposed method

• Utilizes the canonical left brace structure on the structure group $G(X,r)$, where the additive group is free abelian with basis $X$ and the multiplicative group acts via the left multiplication maps $\mathcal{L}_a$.
• Applies the concept of the socle $\operatorname{Soc}(G)$ as the kernel of the homomorphism $\mathcal{L}: G \to \operatorname{Sym}_X$, which identifies elements acting trivially on $X$.
• Constructs an ideal $I = \{ng \mid g \in G(X,r)\}$ in the structure group $G(X,r)$ for $n = [G : \operatorname{Soc}(G)]$, showing $I$ is invariant under the multiplicative group and the left action maps.
• Uses the quotient $G/I$ to construct a finite left brace of order $n^m$, where $m = |X|$, and proves the natural map $X \to G/I$ is injective, embedding $X$ into the brace.
• Applies results from Jacobson radical rings and adjoint groups, leveraging the equivalence between two-sided braces and radical rings to analyze group-theoretic properties.
• Employs the radical chain $A^{(n+1)} = A^{(n)} * A$ in a left brace $A$ to relate the multipermutation level of the solution to the nilpotency structure of the brace.

Experimental results

Research questions

• RQ1 Can the structure group $G(X,r)$ of a finite non-trivial solution $(X,r)$ of the Yang-Baxter equation be an Engel group?
• RQ2 What is the relationship between the multipermutation level of a solution and the radical chain $A^{(n)}$ in the associated left brace $A$?
• RQ3 How can finite solutions be embedded into finite braces and finite braided groups?
• RQ4 What are the group-theoretic properties of structure groups of multipermutation solutions, particularly in relation to being poly-ℤ groups?
• RQ5 Is there a structural obstruction preventing $G(X,r)$ from being an Engel group, and if so, what does it reveal about the algebraic nature of such solutions?

Key findings

• The structure group $G(X,r)$ of any finite non-trivial solution $(X,r)$ of the Yang-Baxter equation cannot be an Engel group.
• Finite multipermutation solutions yield structure groups that are poly-ℤ groups, but these are not Engel groups, providing a rich source of such examples.
• Every finite solution $(X,r)$ can be embedded into a finite left brace via the quotient $G(X,r)/I$, where $I$ is the ideal generated by $nG$ for $n = [G : \operatorname{Soc}(G)]$.
• The quotient $G(X,r)/I$ is a finite left brace of order $n^m$, where $m = |X|$, and the image of $X$ in this quotient is isomorphic to $X$, ensuring the embedding is injective.
• The multipermutation level of a solution is directly related to the length of the radical chain $A^{(n)}$ in the associated left brace $A$, as defined by Rump.
• The structure group $G(X,r)$ inherits a canonical left brace structure from the free abelian group on $X$ and the action of $\mathcal{L}_a$, making it a left brace with $\mathcal{L}_a$ being automorphisms of the additive group.