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Uniqueness of addition in Lie rings $\mathfrak{gl}_n(K)$ and $\mathfrak{sl}_n(K)$

This paper proves that commutator-preserving bijections from $\mathfrak{gl}_n(K)$ to any Lie ring are additive on $\mathfrak{sl}_n(K)$ , thereby establishing criteria for the uniqueness of addition in these Lie rings and demonstrating the additivity of commutator-preserving injections for $\mathfrak{sl}_2(K)$ .

Original authors: Gennadiy Sosnov

Published 2026-06-09

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Gennadiy Sosnov

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 have a magical box of numbers called a Lie Ring. Inside this box, there are two main ways to play with the numbers:

Adding them together (like putting apples in a basket).
Swapping them (a special operation called a "commutator" or "bracket" that measures how much two numbers fail to commute, or how much they "fight" when you switch their order).

Usually, in math, we assume that if you know how to swap things, you automatically know how to add them. But what if someone gives you a new box of numbers and says, "I promise I've kept the swapping rules exactly the same, but I might have changed the way you add them"?

This paper asks: Is it possible to keep the swapping rules perfect while secretly changing the addition rules?

The author, Gennadiy Sosnov, investigates this question for two specific types of number boxes:

$gl_n(K)$ : The box of all $n \times n$ matrices (grids of numbers).
$sl_n(K)$ : A special subset of those matrices where the numbers on the diagonal add up to zero.

Here is the breakdown of his findings, using simple analogies.

The Main Characters: The "Swappers"

In this story, the author looks at a "magic translator" (a map called $\alpha$ or $\beta$ ). This translator takes a number from the original box and puts it in a new box.

The Rule: The translator must preserve the "swapping" rules. If $A$ and $B$ swap to create $C$ in the old box, their translated versions must swap to create the translated $C$ in the new box.
The Question: Does this translator also have to preserve addition? (i.e., If $A + B = C$ in the old box, must $\text{Translator}(A) + \text{Translator}(B) = \text{Translator}(C)$ in the new box?)

If the answer is "Yes, always," we call the box a "Unique Addition Ring." It means the addition rule is so tightly locked to the swapping rule that you can't change one without breaking the other.

The Findings: When is the Addition "Unique"?

The paper discovers that the answer depends entirely on how big the grid is ( $n$ ) and what kind of numbers are inside the field ( $K$ ).

1. The General Grid ( $gl_n(K)$ )

Think of this as a full grid of numbers.

The Bad News: For almost all grids, the addition rule is NOT unique. You can have a translator that keeps the swapping perfect but messes up the addition.
The One Exception: The only time the addition is forced to be unique is if the grid is odd-sized (like $3 \times 3$ $3 \times 3$ , $5 \times 5$ $5 \times 5$ ) AND the numbers inside are only 0 and 1 (the field with 2 elements, $F_2$ $F_{2}$ ).
- Analogy: Imagine a puzzle. For almost any puzzle size, you can rearrange the pieces to keep the picture looking right (swapping) but change how the pieces fit together (adding). But if the puzzle is odd-sized and made of only black and white tiles, the pieces are so constrained that you can't change the fit without ruining the picture.

2. The Special Grid ( $sl_n(K)$ )

This is a grid where the diagonal numbers sum to zero. It's a tighter, more restricted box.

The Bad News: If the grid is $2 \times 2$ and the numbers are from a field with characteristic 2 (where $1 + 1 = 0$ ), the addition is NOT unique. You can trick the translator.
The Good News: In every other case (if the grid is bigger than $2 \times 2$ $2 \times 2$ , or if the numbers aren't in that specific "characteristic 2" world), the addition IS unique.
- Analogy: The $2 \times 2$ grid in this specific world is like a wobbly table; you can shake it (change addition) without it falling over (breaking swapping). But any larger table, or a table in a different world, is rock solid. If you try to shake it, it breaks.

The New Discovery: The "One-Way" Translator

The paper also introduces a new type of translator: an injection.

Imagine a translator that takes numbers from a small box and puts them into a larger box, but doesn't necessarily fill the whole larger box. It's a "one-way" trip.
The Breakthrough: The author proves that for the specific case of the $2 \times 2$ special grid ( $sl_2$ ) with standard numbers (not characteristic 2), even this one-way translator must preserve addition.
Why this matters: Before this paper, mathematicians knew this was true for "two-way" translators (bijections). This is the first time it has been proven for "one-way" translators. It's like discovering that even if you only send a few items through a security checkpoint, the rules of the checkpoint are so strict that you can't sneak in a fake addition rule.

Summary of the "Rules of the Game"

The paper provides a checklist for mathematicians to know if a Lie Ring has "Unique Addition":

Is it a full grid ( $gl_n$ )?
- If $n$ is even? No.
- If $n$ is odd but the numbers aren't just 0 and 1? No.
- If $n$ is odd AND numbers are only 0 and 1? Yes!
Is it a special grid ( $sl_n$ )?
- If it's $2 \times 2$ and numbers are in the "0+1=0" world? No.
- Otherwise? Yes!

The "Why" (The Secret Sauce)

How did the author prove this? He looked at the "Defect."

Imagine the translator makes a tiny mistake when adding two numbers. Let's call this mistake the "Defect."
The author showed that for these specific grids, the "Defect" is trapped in a tiny, isolated corner of the math world (the center of the ring).
Because the grids are so interconnected (you can build any number by "swapping" two others), if the defect is trapped in that tiny corner, it gets crushed to zero.
If the defect is zero, the addition is perfect.

In short: The paper proves that for most matrix Lie rings, the rules of "swapping" are so powerful that they force the rules of "adding" to stay exactly the same. The only times you can break this lock are in very specific, small, or "even-numbered" mathematical worlds.

Technical Summary: Uniqueness of Addition in Lie Rings $\mathfrak{gl}_n(K)$ and $\mathfrak{sl}_n(K)$

Problem Statement
The paper addresses the "uniqueness of addition" problem for Lie rings, specifically the matrix Lie algebras $\mathfrak{gl}_n(K)$ and $\mathfrak{sl}_n(K)$ over an arbitrary field $K$ . A Lie ring $R$ is defined as a Unique Addition (UA) Lie ring if every commutator-preserving bijection $\alpha: R \to S$ (where $S$ is another Lie ring) is necessarily additive. In other words, the condition $\alpha([a, b]) = [\alpha(a), \alpha(b)]$ for all $a, b \in R$ must imply $\alpha(a + b) = \alpha(a) + \alpha(b)$ .

The author investigates whether this property holds for general linear and special linear Lie algebras, extending previous results that were limited to specific fields (e.g., characteristic zero) or specific dimensions. The paper also introduces the concept of iUA-Lie rings, where the condition applies to commutator-preserving injections rather than just bijections.

Methodology
The core methodology relies on the analysis of the defect of a commutator-preserving map. For a map $\alpha: R \to S$ , the defect $\Delta: R \times R \to S$ is defined as:
$\Delta(a, b) = \alpha(a + b) - \alpha(a) - \alpha(b)$
The goal is to prove that $\Delta(a, b) = 0$ for all $a, b$ .

The paper employs the following technical tools:

Centralizers and Centers: The author utilizes the properties of centralizers $C(a) = \{b \in R \mid [a, b] = 0\}$ and the center $Z(R)$ . Key lemmas establish that the defect $\Delta(a, b)$ lies within the image of the intersection of the centers of the centralizers of $a$ and $b$ (Lemma 2.4).
Structural Decomposition: The proofs involve decomposing matrices into standard bases (matrix units $E_{ij}$ and diagonal elements) and analyzing the defect on these components.
Commutator Width: The paper leverages the fact that elements in $\mathfrak{sl}_n(K)$ (for $n \geq 3$ ) can be expressed as commutators of other elements in the algebra (bracket width 1). This allows the extension of additivity from specific elements to the whole algebra via the Jacobi identity and properties of the defect (Lemma 2.7).
Case Analysis by Field and Dimension: The proofs are divided based on the characteristic of the field $K$ (specifically distinguishing $\text{char } K = 2$ ) and the dimension $n$ (distinguishing $n=2$ from $n \geq 3$ ).

Key Contributions and Results

The paper establishes three main theorems:

1. Uniqueness of Addition for $\mathfrak{gl}_n(K)$ (Theorem A)
The ring $\mathfrak{gl}_n(K)$ is a UA-ring if and only if $n$ is odd and $K = \mathbb{F}_2$ (the field with two elements).

Negative Cases: If $K \neq \mathbb{F}_2$ , non-additive commutator-preserving bijections can be constructed using non-additive field bijections on the scalar part. If $K = \mathbb{F}_2$ and $n$ is even, a specific swap of elements with non-zero trace demonstrates non-additivity.
Positive Case: For $n$ odd and $K = \mathbb{F}_2$ , the center of the algebra is small enough ( $|Z(S)|=2$ ) that the defect must vanish, forcing additivity.

2. Uniqueness of Addition for $\mathfrak{sl}_n(K)$ (Theorem B)
The ring $\mathfrak{sl}_n(K)$ is not a UA-ring if and only if $\text{char } K = 2$ and $n = 2$ .

In all other cases (including $n \geq 3$ for any field, and $n=2$ for $\text{char } K \neq 2$ ), $\mathfrak{sl}_n(K)$ is a UA-ring.
The proof for $n \geq 3$ relies on showing that the defect lies in the center and then vanishing due to the bracket width property of $\mathfrak{sl}_n(K)$ .

3. Additivity of Commutator-Preserving Injections (Theorem C)
Let $K$ be a field with $\text{char } K \neq 2$ . Any commutator-preserving injection $\beta: \mathfrak{sl}_2(K) \to S$ is additive.

This result confirms a specific case of the conjecture that every semisimple Lie algebra is an iUA-Lie ring.
The proof involves a detailed analysis of the standard basis $\{e, f, h\}$ of $\mathfrak{sl}_2(K)$ , showing that the defect vanishes on all linear combinations of basis elements by constructing specific commutator identities (Lemmas 4.5–4.11).
The paper notes that this is the first result establishing the additivity of commutator-preserving injections (as opposed to bijections).

Significance and Claims
The author claims that this work provides a complete characterization of the uniqueness of addition for the fundamental Lie algebras $\mathfrak{gl}_n(K)$ and $\mathfrak{sl}_n(K)$ over arbitrary fields.

It resolves an open question posed in previous literature (specifically [4, Problem 1]) regarding the UA-property of $\mathfrak{gl}_n(K)$ .
It extends the theory of unique addition from associative rings and specific Lie algebras (like those over algebraically closed fields of characteristic zero) to the general case of matrix Lie algebras over arbitrary fields.
The result on injections (Theorem C) is highlighted as a novel contribution, as prior results on uniqueness of addition generally required the map to be a bijection.

The paper concludes by noting that while $\mathfrak{sl}_2(K)$ is an iUA-ring for $\text{char } K \neq 2$ , the situation for $\mathfrak{sl}_2(K)$ in characteristic 2 is different (it is not a UA-ring), and the question of whether the map $\phi(a,b,c) = ([a,[b,c]], [b,[c,a]])$ is surjective for $\mathfrak{gl}_n(K)$ with $n \geq 3$ remains an open problem (Problem 4.12).