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Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra

This paper classifies non-nilpotent non-Lie Leibniz algebras with a one-dimensional derived subalgebra over any field of characteristic not equal to 2 as direct sums of a specific two-dimensional algebra and an abelian algebra, and subsequently determines their derivations, automorphisms, biderivations, and solutions to the coquecigrue problem.

Original authors: Alfonso Di Bartolo, Gianmarco La Rosa, Manuel Mancini

Published 2026-05-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alfonso Di Bartolo, Gianmarco La Rosa, Manuel Mancini

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 an architect trying to understand the blueprints of a very specific, strange type of building. In the world of mathematics, these "buildings" are called Leibniz algebras. They are like the famous "Lie algebras" (which describe symmetries in physics and geometry), but they are a bit more flexible and less symmetrical.

This paper is about a specific, somewhat messy type of these buildings: ones that are not "nilpotent" (meaning they don't collapse into nothingness if you keep applying their rules) and have a very small "core" of activity. Specifically, the authors are looking at buildings where the "derived subalgebra" (the part generated by all the interactions between the building's parts) is just one-dimensional. Think of this as a building where, no matter how many people interact, they only ever produce one specific type of "energy" or "output."

Here is the breakdown of what the authors found, using simple analogies:

1. The Big Discovery: The "Lego Block" Structure

The main result of the paper is a classification. The authors asked: "If we have a non-nilpotent Leibniz algebra with this tiny one-dimensional core, what does it actually look like?"

They found that these complex structures aren't actually unique or mysterious at all. They are always built from two simple pieces stuck together:

Piece A: A tiny, two-dimensional "engine" called $S_2$ . This is the only part that does any real work. It has two parts, let's call them $e_1$ and $e_2$ . The rule is simple: if you combine them in a specific order ( $e_2$ acting on $e_1$ ), you get $e_1$ back. It's like a machine that takes a gear and a lever, and just spins the gear.
Piece B: A large, empty, "boring" room called an abelian algebra. This is just a collection of extra dimensions that sit there and do nothing. They don't interact with each other or the engine in a way that creates new things.

The Conclusion: Any such algebra is just the $S_2$ engine sitting next to a boring empty room. The authors call this whole structure $L_n$ (where $n$ is the total size).

Why this matters: Before this paper, we knew this was true if the math was done with complex numbers (like in the paper [11] they reference). This paper proves it works for any field (like real numbers or others), as long as the number 2 isn't equal to 0 (which is true for almost all standard math). It's like proving a rule about Lego bricks works whether you are building in a vacuum or underwater, as long as the bricks don't dissolve.

2. The "Tools" for the Building

Once they identified the structure ( $L_n$ ), the authors built a "toolkit" to analyze it. They calculated three specific things:

Derivations (The Repair Crew): These are ways you can tweak the building's rules without breaking the structure. The authors wrote down a specific matrix (a grid of numbers) that shows exactly what these tweaks look like. It's like having a manual that says, "You can change the speed of the engine, or add a few extra people to the empty room, but you can't change the core engine's fundamental rule."
Automorphisms (The Renovation Team): These are ways to rearrange the building that make it look different but act exactly the same. The authors found the specific shape of these rearrangements.
Biderivations (The Double-Check System): This is a more advanced concept involving pairs of tools that check the building from two angles at once. They calculated exactly what these pairs look like for this specific building.

3. Solving the "Coquecigrue" Problem

This is the most abstract part, but here is the simple version:

In math, there is a famous rule called the "Lie Third Theorem." It says that for every Lie algebra (a type of symmetry), there is a corresponding "Lie Group" (a smooth, continuous shape, like a sphere or a donut) that you can "integrate" or build from it.

For Leibniz algebras, this rule was broken. Mathematicians (like J.-L. Loday) asked: "Is there a similar shape for Leibniz algebras?" This question is called the Coquecigrue problem (named after a mythical creature, implying it's a bit elusive).

The authors solved this for their specific building ( $L_n$ ).

They found a shape called a Lie Rack.
The Analogy: Imagine a Lie Group is a smooth, round ball. A Lie Rack is a slightly weird, lopsided shape. If you zoom in really close to the center of this lopsided shape and look at its "slope" (the tangent space), it perfectly matches the rules of their Leibniz algebra ( $L_n$ ).
They explicitly wrote down the formula for how to move around on this shape. It involves a simple exponential function ( $e^{x_1}$ ), which acts like a "boost" that changes the second coordinate based on the first.

Summary

In plain English, this paper says:

We found the pattern: Any non-nilpotent Leibniz algebra with a one-dimensional core is just a tiny, active 2D engine ( $S_2$ ) attached to a bunch of inactive, empty space.
We mapped the rules: We wrote down the exact mathematical formulas for how to tweak, rearrange, and double-check this structure.
We built the shape: We solved a long-standing puzzle by showing exactly what the "smooth shape" (Lie Rack) looks like that corresponds to this algebra, proving that even these weird, non-symmetrical algebras have a geometric home.

The paper is a piece of pure structural mathematics: it takes a specific, slightly messy category of objects, proves they are all built from the same simple Lego blocks, and then builds the geometric "house" that fits inside them.

Problem Statement
The paper addresses the classification and structural analysis of non-nilpotent, non-Lie Leibniz algebras over a field $F$ (where $\text{char}(F) \neq 2$ ) that possess a one-dimensional derived subalgebra. While the classification of nilpotent Leibniz algebras with a one-dimensional derived subalgebra was previously established (identifying Heisenberg, Kronecker, and Dieudonné algebras), the non-nilpotent case remained less understood, particularly over general fields. Specifically, the authors aim to generalize a result by [11], which proved that complex non-split non-nilpotent non-Lie Leibniz algebras with these properties are isomorphic to a specific two-dimensional algebra, to the context of arbitrary fields with characteristic not equal to 2. Additionally, the paper seeks to characterize the derivations, automorphisms, and biderivations of these algebras and solve the "coquecigrue problem" (integrating the algebra into a Lie rack) for this specific class.

Methodology
The authors employ the theory of non-abelian extensions of Leibniz algebras, as introduced in [18]. The core methodological approach involves:

Decomposition: Identifying a two-dimensional bilateral ideal $S$ within the algebra $L$ . Using the classification of two-dimensional Leibniz algebras by Cuvier [8], they determine that $S$ must be isomorphic to either a Lie algebra ( $S_1$ ) or a non-Lie Leibniz algebra ( $S_2$ ).
Extension Analysis: Treating $L$ as an extension of an abelian algebra $L_0 \cong F^{n-2}$ by $S$ . The authors analyze the associated 2-cocycle $\omega$ and the actions $l, r$ of $L_0$ on $S$ by solving the system of equations (L1)–(L7) that govern Leibniz algebra extensions.
Case Differentiation:
- Case 1 ( $S \cong S_1$ ): They demonstrate that the resulting extension yields a Lie algebra, which decomposes into a direct sum of $S_1$ and the abelian part.
- Case 2 ( $S \cong S_2$ ): They analyze the biderivations of $S_2$ and the constraints on the 2-cocycle. Through basis changes and algebraic manipulation, they show that the extension splits, resulting in a direct sum.
Structural Computation: For the resulting algebra $L_n$ , the authors explicitly compute the matrix forms for the Lie algebra of derivations ( $\text{Der}(L_n)$ ), the Lie group of automorphisms ( $\text{Aut}(L_n)$ ), and the Leibniz algebra of biderivations ( $\text{Bider}(L_n)$ ).
Integration: To address the coquecigrue problem, the authors utilize M. K. Kinyon's results on split Leibniz algebras and Lie racks [15] to construct an explicit integration of $L_n$ into a Lie rack over $\mathbb{R}^n$ .

Key Contributions and Results

Classification Theorem (Theorem 2.2): The primary result establishes that any non-nilpotent, non-Lie left Leibniz algebra $L$ over a field $F$ ( $\text{char}(F) \neq 2$ ) with $\dim_F L = n$ and $\dim_F [L, L] = 1$ is isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra $S_2$ (defined by basis $\{e_1, e_2\}$ and bracket $[e_2, e_1] = e_1$ ) and an abelian algebra of dimension $n-2$ . This algebra is denoted as $L_n$ .
Generalization of Previous Work: This result generalizes Theorem 2.6 from [11], which was restricted to the complex field ( $F=\mathbb{C}$ ), to any field with characteristic not equal to 2.
Corollary on Non-Split Algebras: The paper proves that if such an algebra is non-split (indecomposable), then its dimension must be $n=2$ , and it is isomorphic to $S_2$ .
Structural Characterization: The authors provide explicit matrix representations for:
- The Lie algebra of derivations $\text{Der}(L_n)$ .
- The Lie group of automorphisms $\text{Aut}(L_n)$ .
- The Leibniz algebra of biderivations $\text{Bider}(L_n)$ .
Solution to the Coquecigrue Problem: The paper explicitly constructs a Lie rack structure on the vector space $\mathbb{R}^n$ whose tangent space at the identity is isomorphic to $L_n$ . The binary operation is defined as:
$(x_1, \dots, x_n) \rhd (y_1, \dots, y_n) = (y_1, y_2 + e^{x_1}y_2, y_3, \dots, y_n)$
This completes the classification of Lie racks whose tangent spaces yield Leibniz algebras with one-dimensional derived subalgebras.

Significance
The paper claims to complete the classification of Leibniz algebras with one-dimensional derived subalgebras by addressing the non-nilpotent case. By generalizing results from the complex field to arbitrary fields of characteristic $\neq 2$ , it provides a more robust structural understanding of these algebras. Furthermore, by explicitly solving the coquecigrue problem for this class, the work bridges the gap between the algebraic structure of non-Lie Leibniz algebras and the geometric theory of Lie racks, demonstrating how these specific non-nilpotent algebras can be integrated into smooth geometric objects. The detailed computation of derivations and automorphisms offers a concrete foundation for further study of the symmetry and deformation properties of these algebras.

1. The Big Discovery: The "Lego Block" Structure

2. The "Tools" for the Building

3. Solving the "Coquecigrue" Problem

Summary

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