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Non-isomorphic restricted Lie algebras with isomorphic restricted enveloping algebras

This paper demonstrates that the restricted isomorphism problem has a negative answer over any field of positive characteristic by constructing, for every dimension at least $p+5$ , pairs of non-isomorphic finite-dimensional $p$ -nilpotent restricted Lie algebras that possess isomorphic restricted enveloping algebras.

Original authors: Xabier García-Martínez

Published 2026-06-17

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Xabier García-Martínez

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 a detective trying to identify a suspect. Usually, if you have a perfect fingerprint (a unique set of features), you can identify exactly who the person is. In the world of mathematics, specifically in a field called Lie algebras, mathematicians have long believed that if two structures produce the same "fingerprint" (their restricted enveloping algebras), they must be the same structure underneath.

This paper, written by Xabier García-Martínez, proves that this belief is wrong. The author shows that you can have two completely different mathematical structures that produce the exact same fingerprint, making them indistinguishable to the tools we usually use.

Here is a simple breakdown of the story:

1. The Setup: The "Fingerprint" vs. The "Person"

Think of a Restricted Lie Algebra as a complex machine built from gears and levers. It has specific rules about how the parts move and interact.

The Machine (The Lie Algebra): This is the actual object. It has a specific shape, size, and internal complexity.
The Fingerprint (The Restricted Enveloping Algebra): This is a mathematical "shadow" or a "blueprint" generated by the machine. It records how the machine behaves when you run it through a specific test.

For a long time, mathematicians asked: "If two machines produce the exact same blueprint, are they the same machine?"
The answer, according to this paper, is No.

2. The Counter-Example: The Twin Machines

The author builds two different machines, let's call them Machine L and Machine H.

They are built in a world where numbers behave differently than in our everyday math (specifically, in a world with a "prime" number characteristic, like counting in cycles of 2, 3, 5, etc.).
Machine L is slightly simpler inside. If you count its internal "tangled gears" (mathematicians call this the derived subalgebra), there are exactly $p$ of them.
Machine H is slightly more complex inside. It has $p + 1$ tangled gears.

Because their internal complexity is different, Machine L and Machine H are definitely not the same. You could take them apart and see the difference immediately.

3. The Twist: The Identical Blueprints

Here is the magic trick. When the author runs both machines through the "fingerprint test" (creating the restricted enveloping algebra), the results are identical.

The blueprint for Machine L looks exactly like the blueprint for Machine H.
If you only looked at the blueprint, you would swear they were the same machine.

The paper proves that this happens for machines of a certain size (specifically, when the machine is large enough, at least $p + 5$ parts).

4. Why This Matters (The "So What?")

In the past, mathematicians had found similar tricks with groups (another type of mathematical structure), but they were unsure if it worked for these specific "restricted" Lie algebras, especially ones that are "p-nilpotent" (a fancy way of saying they eventually stop moving if you keep applying a specific rule).

This paper settles the debate. It says:

The "Restricted Isomorphism Problem" has a negative answer.
You cannot always tell two different restricted Lie algebras apart just by looking at their restricted enveloping algebras.
Even if you know the "blueprint" perfectly, you might still be missing the true identity of the machine because the blueprint doesn't capture the exact number of internal tangled gears.

5. How They Found It

The author didn't just guess. They used a computer to search through millions of possibilities.

Imagine trying to find two different keys that open the same lock. The computer tried thousands of key shapes.
It found a pair where the keys looked different (different numbers of teeth/gears), but when you tried them in the lock (the algebra test), they both turned perfectly and opened it in the exact same way.
Once the computer found a working pair in a simple case (using the number 2), the author figured out the mathematical rules to build these pairs for any prime number (3, 5, 7, etc.).

Summary Analogy

Imagine two different cars:

Car A has 4 cylinders.
Car B has 5 cylinders.

They are clearly different cars. However, the author discovered a way to build them so that if you take a photo of their exhaust fumes (the "enveloping algebra"), the photos are pixel-perfect identical. You cannot tell Car A from Car B just by looking at the exhaust.

This paper proves that in the world of these specific mathematical structures, the "exhaust photo" is not enough to identify the car. You need to look under the hood to see the real difference.

Technical Summary: Non-Isomorphic Restricted Lie Algebras with Isomorphic Restricted Enveloping Algebras

Problem Statement
The paper addresses the Restricted Isomorphism Problem, a modular analogue of the classical Modular Isomorphism Problem for group algebras. The problem asks whether a finite-dimensional restricted Lie algebra $L$ over a field $F$ of characteristic $p > 0$ is uniquely determined by its restricted enveloping algebra $u(L)$ . Specifically, if $u(L) \cong u(H)$ as $F$ -algebras, must $L \cong H$ as restricted Lie algebras?

While the problem has been settled positively for specific classes (e.g., abelian, metacyclic, or $p$ -nilpotent algebras over perfect fields in certain dimensions) and negatively for ordinary universal enveloping algebras $U(g)$ in prime characteristic, the general case for restricted Lie algebras remained open. The author notes that previous positive results for $p$ -nilpotent algebras suggested a potential affirmative answer for this class, particularly over perfect fields.

Methodology
The author constructs explicit counterexamples to the Restricted Isomorphism Problem over fields of arbitrary positive characteristic. The methodology involves:

Case-by-Case Construction:
- Characteristic 2: A parametric family of algebras $L_k$ and $H_k$ is constructed over $\mathbb{F}_2$ . These algebras are defined by specific bases, $p$ -maps (2-maps), and non-zero brackets. The construction relies on the specific identities of characteristic 2, such as $(x+y)^2 = x^2 + y^2 + [x,y]$ .
- Odd Characteristic: A distinct family $L_{p,k}$ and $H_{p,k}$ is constructed over $\mathbb{F}_p$ for odd primes $p$ . This construction adapts the mechanism of the characteristic 2 case but requires different bracket definitions and utilizes the invertibility of 2 to define specific central elements.
Isomorphism Verification: For each pair, the author defines a map $\phi$ on the generators of $u(L)$ to words in the generators of $u(H)$ . It is rigorously verified that these images satisfy the defining relations (commutators and $p$ -power relations) of $u(L)$ , thereby establishing a well-defined algebra homomorphism. Surjectivity is proven by showing the generators of $u(H)$ lie in the image, and injectivity follows from the equality of dimensions (guaranteed by the Poincaré–Birkhoff–Witt theorem).
Generalization: Using the property that restricted enveloping algebras commute with scalar extensions ( $u(L \otimes_{\mathbb{F}_p} F) \cong u(L) \otimes_{\mathbb{F}_p} F$ ), the results are extended from prime fields to any field $F$ of characteristic $p$ .
Discovery Process: The author notes that the initial examples were discovered via a computer search guided by "determined invariants" (such as dimension-subalgebra quotients), specifically looking for pairs where the derived subalgebra dimensions differed ( $\dim L' \neq \dim H'$ ) while other graded invariants matched.

Key Contributions and Results
The primary result is Theorem 5.2, which states:
Let $F$ be a field of characteristic $p > 0$ . For every integer $n \geq p + 5$ , there exist $p$ -nilpotent restricted Lie algebras $L$ and $H$ over $F$ such that:

$\dim L = \dim H = n$ .
$u(L) \cong u(H)$ as $F$ -algebras.
$\dim L' = p$ and $\dim H' = p + 1$ .

Consequently, $L$ and $H$ are not isomorphic as restricted Lie algebras (and not even as ordinary Lie algebras) because their derived subalgebras have different dimensions.

Significance and Claims
The paper claims to provide a negative solution to the Restricted Isomorphism Problem over every field of positive characteristic. The significance of this result is threefold:

Resolution of Open Problems: It answers negatively Problem 6 (existence of a counterexample) and Problem 9 (whether the answer is positive for $p$ -nilpotent algebras over perfect fields) from Usefi's survey [17].
Limit of Determinacy: The results demonstrate that the restricted enveloping algebra $u(L)$ does not determine the dimension of the derived subalgebra $L'$ , nor the lower central series of $L$ , even within the class of $p$ -nilpotent algebras. This establishes a sharp boundary for the invariants captured by $u(L)$ ; while it determines the associated graded restricted Lie algebra $\text{gr } L$ and quotients like $L/(L'^p + \gamma_3(L))$ , it fails to capture the dimension of the derived subalgebra.
Distinction from Ordinary Enveloping Algebras: The paper clarifies that the phenomenon is specific to the restricted setting. For the ordinary universal enveloping algebra $U(g)$ , the isomorphism $U(L) \cong U(H)$ would imply $\dim(L/L') = \dim(H/H')$ . Since the constructed pairs have $\dim(L/L') \neq \dim(H/H')$ , it follows that $U(L) \not\cong U(H)$ . Thus, the counterexamples rely crucially on the collapse of the $p$ -closure of the derived subalgebra in the commutative quotient of the restricted enveloping algebra.

The author concludes that the restricted isomorphism problem has a negative answer even for $p$ -nilpotent algebras over perfect fields, settling a long-standing open question in the representation theory of restricted Lie algebras.