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Addendum/Corrigendum to "On solubility of skew left braces and solutions of the Yang-Baxter equation"

This paper corrects a proof in a previous study regarding the solubility of skew left braces and Yang-Baxter equation solutions by introducing "i-homomorphisms," which refine the definition of solution solubility and confirm that a skew brace is soluble if and only if its associated solution is soluble.

Original authors: A. Ballester-Bolinches, R. Esteban-Romero, P. Jiménez-Seral, V. Pérez-Calabuig

Published 2026-04-24

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CC BY 4.0

Original authors: A. Ballester-Bolinches, R. Esteban-Romero, P. Jiménez-Seral, V. Pérez-Calabuig

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a massive, intricate puzzle called the Yang-Baxter Equation. In the world of mathematics, this isn't just a puzzle; it's a fundamental rule that describes how particles interact in physics and how information flows in cryptography.

To solve this puzzle, mathematicians use a special tool called a Skew Left Brace. Think of a Skew Left Brace as a "machine" or a "factory" that takes a set of items and rearranges them according to specific rules.

The Problem: A Broken Blueprint

In a previous paper (published in 2024), the authors claimed they had found a perfect way to link these "machines" (Skew Braces) to the "puzzle pieces" (Solutions). They said:

"If the machine is 'soluble' (meaning it can be broken down into simple, manageable parts), then the puzzle piece it creates is also 'soluble' (meaning it can be broken down too)."

They called this Theorem C. It was a big deal because it promised a shortcut: if you understand the machine, you automatically understand the puzzle.

However, there was a glitch.
The authors realized they missed a tiny, crucial step in their logic. It was like building a bridge and forgetting to check if the bolts were tightened correctly. Because of this missing step, their proof that "Machine = Puzzle" was shaky. Some "puzzle pieces" that looked simple were actually complex, and vice versa.

The Fix: Introducing the "i-Homomorphism"

To fix the bridge, the authors invented a new tool called an i-homomorphism.

Here is a simple analogy to understand what this new tool does:

Imagine you have a giant, messy room full of people (the Solution).

Old Method: You tried to group people by asking, "Who looks the same?" If two people looked the same, you put them in the same box. But sometimes, this grouping was too loose. You might put two very different people in the same box just because they wore the same color shirt, missing the fact that they had different personalities.
New Method (i-homomorphism): The authors introduced a stricter rule. Now, to group people, you don't just look at their shirts. You look at how they interact with a specific "Key Person" in the room.
- If a group of people interacts with the "Key Person" in a specific, predictable way (swapping places perfectly), then they can be grouped together.
- This new grouping is called an i-kernel.

This new rule is special because it perfectly matches the "structure" of the machine (the Skew Brace) with the "structure" of the puzzle.

The Big Discovery: Simplicity vs. Complexity

With this new tool, the authors redefined what it means for a solution to be Soluble.

Indecomposable (The "Unbreakable" Puzzle): Imagine a puzzle piece that is so tightly woven that you cannot split it into two smaller, independent pieces. In the old view, this was hard to identify. In the new view, if a puzzle is "i-simple" (it has no valid "Key Person" groups to split it), it is Indecomposable. It's a solid block.
Soluble (The "Layer Cake" Puzzle): A soluble solution is like a layer cake. You can slice it layer by layer. Each slice is a simpler version of the cake, until you get down to a single crumb.

The Results: The Bridge is Fixed!

By using this new "Key Person" grouping method, the authors repaired their bridge:

The Machine and the Puzzle Match Again: They proved that if the "Machine" (Skew Brace) is a "Layer Cake" (Soluble), then the "Puzzle" it creates is also a "Layer Cake." The original claim (Theorem C) is now true and solid.
New Insights: They found that some puzzles that look like they should be breakable (because the machine is simple) are actually unbreakable blocks. The new tool helps them spot these tricky cases.
A Safety Net: They also found a specific condition (if a certain "Key Person" acts like a neutral element) that guarantees a puzzle is breakable, even if the machine is complex.

Summary

Think of this paper as a team of architects who built a bridge between two islands (Math and Physics). They realized they used the wrong blueprint, so the bridge wobbled. They didn't just patch the hole; they redesigned the entire support system using a new, stronger material (the i-homomorphism).

Now, the bridge is stable. If you know the structure of the "Machine" island, you can confidently walk across to the "Puzzle" island and know exactly how the pieces fit together. This helps scientists and mathematicians better understand the fundamental rules of how things interact in our universe.

1. Problem Statement

The paper addresses a critical gap identified in the authors' previous work (Adv. Math. 455, 2024), specifically regarding Theorem C. In the original paper, the authors claimed that a skew brace is soluble if and only if its associated solution to the Yang-Baxter Equation (YBE) is soluble.

However, a review by the authors (prompted by external feedback) revealed an "irreparable mistake" in the proof of Theorem C. The error stemmed from an oversight in the definition of homomorphisms of solutions. The original definition was insufficient to guarantee that the structural properties of the solution (specifically solubility) correctly mapped to the properties of the associated structure skew brace ( $G_X$ ). Consequently, the equivalence between the solubility of the solution and the solubility of the skew brace was not rigorously established under the original definitions.

The primary goal of this addendum is to repair the proof of Theorem C by introducing a refined class of homomorphisms and redefining the concept of solubility for solutions.

2. Methodology and New Definitions

To resolve the inconsistency, the authors introduce a new class of mappings called i-homomorphisms (ideal-homomorphisms) and redefine indecomposability and solubility based on these mappings.

A. i-Homomorphisms and i-Kernels

Definition: A homomorphism of solutions $f: (X, r) \to (Y, s)$ is an i-homomorphism if there exists a specific equivalence class $X_0 \subseteq X$ (called an i-kernel) such that for every equivalence class $Z$ of $\ker f$ , the solution satisfies:
$r(X_0, Z) = Z \times X_0 \quad \text{and} \quad r(Z, X_0) = X_0 \times Z$
Significance: Unlike standard homomorphisms, i-homomorphisms enforce a specific interaction between the kernel and the rest of the solution space. This structure allows the kernel to induce ideals in the associated structure skew brace.

B. i-Simplicity and Indecomposability

i-Simple Solution: A non-singleton solution is i-simple if the only i-kernel of any i-homomorphism is the entire set $X$ .
Key Characterization (Proposition 3.3): The authors prove that a solution is indecomposable if and only if it is i-simple. This corrects the previous understanding where "simple" (in the standard homomorphism sense) did not perfectly align with "indecomposable."

C. Connection to Structure Skew Braces

Theorem 3.9: The authors establish that the ideal generated by an i-kernel $X_0$ in the structure skew brace $G_X$ (denoted $\langle X_0 \rangle$ ) forms a strong left ideal. Furthermore, the ideal generated by the i-kernel and the kernel of the induced map on the structure group, $I_{X_0} = \langle X_0 \rangle \cdot \ker \bar{f}$ , is a full ideal of $G_X$ .
This establishes a rigorous bridge: i-homomorphisms of solutions correspond directly to ideal quotients in the structure skew brace.

D. Redefinition of Soluble Solutions

Definition 4.1: A solution $(X, r)$ is defined as soluble if there exists a chain of i-epimorphisms $f_k: X \to B_k$ with i-kernels $X_k$ such that the images of the previous kernels form abelian ideals in the target skew braces.
This definition positions soluble solutions as the "opposite" of indecomposable (i-simple) solutions, mirroring the definition of soluble skew braces.

3. Key Results and Theorems

Theorem 4.2 (Main Repair)

Statement: If a solution $(X, r)$ is soluble (under the new i-homomorphism definition), then its associated structure skew brace $G_X$ is soluble.

Significance: This repairs the "only if" direction of the original Theorem C, which was previously flawed. The proof relies on lifting the chain of i-epimorphisms to a chain of ideals in $G_X$ .

Corollary 4.3 (Restored Equivalence)

Statement: For a skew brace $B$ , the following are equivalent:

$B$ is a soluble skew brace.
The associated solution $(B, r_B)$ is a soluble solution.
The structure skew brace $G_B$ is soluble.
The permutation skew brace $G(B, r_B)$ is soluble.

Significance: This fully restores the main result of the original paper, confirming that the solubility of the algebraic structure and the solution are equivalent, provided the new definitions are used.

Theorem 4.4 (Sufficient Condition for Converse)

Statement: Let $(X, r)$ be a solution with a soluble structure skew brace $G_X$ . If there exists an element $x \in X$ such that $\lambda_x = \rho_x = \text{id}_X$ (i.e., $x$ is a fixed point for the structure maps), then $(X, r)$ is a soluble solution.

Significance: This provides a sufficient condition for the converse of Theorem 4.2. It generalizes the original Theorem D.

Counter-Example (Example 4.7)

The authors demonstrate that the converse of Theorem 4.2 is not true in general. They present a Lyubashenko solution on 3 elements which has a soluble structure skew brace (it is right nilpotent) but is a simple (and thus non-soluble) solution. This highlights that the condition $\lambda_x = \rho_x = \text{id}_X$ in Theorem 4.4 is essential.

4. Significance and Impact

Correction of Literature: The paper successfully rectifies a fundamental error in the classification of solubility for Yang-Baxter solutions, ensuring that the relationship between skew braces and their solutions is mathematically sound.
Refined Classification Tools: By introducing i-homomorphisms, the authors provide a more robust tool for analyzing indecomposable solutions. The equivalence between indecomposability and i-simplicity offers a clearer path for classifying minimal solutions.
Structural Insight: The work clarifies the relationship between the "twist" nature of solutions and the ideal structure of skew braces. It shows that while simple solutions do not always correspond to simple skew braces, the new i-simple concept does.
Multipermutation Solutions: The results connect soluble solutions to multipermutation solutions. While all multipermutation solutions have soluble structure skew braces, the paper clarifies that they are not necessarily soluble solutions unless specific conditions (like the existence of a fixed point) are met.

Conclusion

This addendum is a crucial correction that strengthens the theoretical framework linking skew left braces and the Yang-Baxter equation. By redefining solubility through the lens of i-homomorphisms, the authors successfully prove that the solubility of a solution is equivalent to the solubility of its associated structure skew brace, provided the solution satisfies specific structural conditions. This resolves the ambiguity in the previous work and establishes a more precise classification for non-degenerate solutions.