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On anti-left-invariant and left-invariant pseudo-Euclidean nearly associative algebras

This paper introduces and classifies anti-left-invariant and left-invariant pseudo-Euclidean nearly associative algebras by establishing their correspondence with Levi-Civita products of Lie and Jordan structures, proving specific nilpotency and solvability properties, and demonstrating that both classes can be recursively constructed via double extension procedures from simpler algebraic foundations.

Original authors: Elisabete Barreiro, Said Benayadi, Hamza El Ouali, Carla Rizzo

Published 2026-08-04

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Elisabete Barreiro, Said Benayadi, Hamza El Ouali, Carla Rizzo

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 the universe not just as a collection of stars and planets, but as a giant, invisible dance floor where every move is governed by strict rules. In the world of mathematics, specifically a branch called geometry, scientists study these rules to understand how shapes bend, twist, and interact. One of the most famous "dance moves" is the Levi-Civita connection, a way of describing how things change as they slide across a curved surface without slipping. Usually, this happens on a stage called a "Lie group," which is a fancy name for a shape that also has a built-in algebra (a set of rules for combining things).

But what if the dance floor itself is a bit weird? What if the rules for combining moves don't follow the standard "associative" law (where the order of grouping doesn't matter, like $(A \times B) \times C = A \times (B \times C)$ )? This is where "nearly associative algebras" come in. They are like a dance where the steps are almost perfectly coordinated, but with a tiny, rhythmic glitch that makes them unique. When you add a "pseudo-Euclidean" metric to this mix, you're essentially giving the dancers a specific way to measure distance and angles, even if that space is a bit strange (like having some directions that act like time and others like space). The big question mathematicians have been asking is: Can we build these weird, glitchy dance floors from scratch? Can we take a simple, boring floor and add layers to it until we get these complex structures? This paper dives deep into that question, trying to map out exactly how these algebraic dance floors are constructed.

The Great Algebraic Construction Project

In this paper, the authors act like master architects and LEGO builders, but instead of plastic bricks, they are using abstract mathematical structures called "nearly associative algebras." They are investigating two very specific types of these algebras: anti-left-invariant and left-invariant pseudo-Euclidean nearly associative algebras.

To understand what they found, imagine you have a set of magical building blocks. The authors discovered that no matter how complex or weird your final structure looks, you can always build it by starting with a very simple, empty block and adding layers one by one using a specific technique they call a "double extension."

The Two Types of Dance Floors

The paper splits its investigation into two main categories, which are like two different styles of dance:

The Anti-Left-Invariant Style:
Think of this as a dance where the partners move in opposite directions relative to the music. The authors proved that if you build a structure using these rules, it has a very strict limit on how "tall" or complex it can get. Specifically, they showed that these algebras are nilpotent of index at most five. In plain English, this means if you keep multiplying the elements together (like stacking blocks), the stack will collapse and become zero after at most five steps. It's a finite game.

They also found that these structures are deeply connected to something called Jacobi-Jordan algebras (a type of algebra that behaves like a mix of Lie and Jordan algebras). They proved that the "underlying" algebra of these structures is always cyclic, meaning it has a specific, repeating pattern.

The Big Surprise: The authors explicitly ruled out a whole class of these algebras. They proved that if you try to build an anti-left-invariant algebra on a "Euclidean" space (a standard, positive-distance space like the room you're sitting in), the result is always trivial. This means the dance floor is empty; nothing happens. The only way to get a non-trivial (interesting) structure is if the space has a "Lorentzian" signature (like in relativity, with time and space mixed) or a specific signature like $(2, n-2)$ . Even then, they found that for signature $(2, n-2)$ , the algebra collapses even faster, becoming nilpotent of index at most three.
The Left-Invariant Style:
This is the dance where the partners move in sync. The authors discovered that these structures are slightly more flexible. They proved that the Lie algebra (the "skeleton" of the structure) associated with these algebras is two-step solvable. This is a fancy way of saying the structure unravels quickly if you try to break it down.

Crucially, they showed that if you are working in a standard Euclidean space (positive distances only), these algebras must be quadratic commutative associative algebras. In other words, if the space is "normal," the "glitch" disappears, and you just get a standard, boring associative algebra. The "nearly" part only survives if the space is weird (pseudo-Euclidean).

The Magic of Double Extension

The core of the paper is the construction method. The authors introduced a procedure called double extension. Imagine you have a small, stable platform (a lower-dimensional algebra). To build a bigger one, you don't just stack a block on top; you add a "shadow" layer and a "core" layer simultaneously. You connect them with specific rules (maps and bilinear forms) that ensure the new, bigger structure still follows the "nearly associative" dance rules.

They proved a powerful theorem: Every non-trivial anti-left-invariant or left-invariant pseudo-Euclidean nearly associative algebra can be built this way. You start with a trivial (empty) algebra or a simple quadratic commutative associative algebra, and you apply a finite sequence of these double extensions until you reach your target.

Classifying the Small Ones

To make sure their theory works, the authors went into the lab and classified all the possible structures of small sizes.

For Anti-Left-Invariant: They listed every possible algebra up to dimension 6 with a specific signature $(2, n-2)$ . They found that these are all built from trivial algebras using their double extension method.
For Left-Invariant: They classified all Lorentzian (space-time like) non-commutative algebras up to dimension 4. They found that dimension 2 is impossible (no such non-commutative algebra exists), dimension 3 has exactly one type, and dimension 4 has exactly two types.

What This Means

The paper doesn't just list examples; it provides a unified framework. It tells us that these complex, geometric-algebraic objects aren't random accidents. They are all constructed from the same simple ingredients using the same recipe. If you want to understand the geometry of these "nearly associative" spaces, you just need to understand how to perform the double extension.

The authors are very sure of their results; they didn't just suggest these patterns or run simulations. They provided rigorous mathematical proofs that these structures must be nilpotent, must be solvable, and must be constructible via double extensions. They effectively closed the door on the idea that there could be some hidden, exotic algebra of this type that doesn't fit their construction method. For anyone trying to build these mathematical worlds, the paper provides the complete blueprint.

Technical Summary: Anti-left-invariant and Left-invariant Pseudo-Euclidean Nearly Associative Algebras

Problem Statement and Motivation
The paper addresses the algebraic structure of pseudo-Euclidean nearly associative algebras arising from Levi-Civita products associated with pseudo-Euclidean Lie and Jordan algebras. While the Levi-Civita connection on pseudo-Riemannian Lie groups is well-understood—yielding left-symmetric algebras when the curvature vanishes—this construction has been extended to broader non-associative settings. The authors focus on two specific classes of pseudo-Euclidean nearly associative algebras:

Anti-left-invariant: Those where the metric satisfies $\langle u \cdot v, w \rangle = -\langle v, u \cdot w \rangle$ . These arise from the Levi-Civita product of pseudo-Euclidean Lie algebras.
Left-invariant: Those where the metric satisfies $\langle u \cdot v, w \rangle = \langle v, u \cdot w \rangle$ . These arise from the Levi-Civita product of pseudo-Euclidean Jordan algebras.

The central problem is to establish the structural properties of these algebras, characterize their underlying Lie and Jordan structures, and provide a recursive classification framework using double extension techniques.

Methodology
The authors employ a combination of algebraic identities, representation theory, and constructive methods:

Algebraic Characterization: The paper utilizes the defining identity of nearly associative algebras, $(u \cdot v) \cdot w = v \cdot (w \cdot u)$ , alongside the specific invariance properties of the bilinear form to derive constraints on multiplication operators and associators.
Structural Analysis: The authors analyze the solvability and nilpotency of the underlying Lie algebras ( $A^-$ ) and Jordan/Jacobi-Jordan algebras ( $A^+$ ). They investigate the properties of the derived algebra $[A, A]$ and its orthogonal complement.
Double Extension Procedure: Motivated by the classical construction of Medina and Revoy for quadratic Lie algebras, the paper introduces a generalized double extension procedure. This involves extending a base algebra by a commutative associative algebra (or its dual) using specific linear maps and bilinear forms that satisfy compatibility conditions derived from the nearly associative and invariance axioms.
Inductive Classification: By proving that non-trivial algebras in these classes contain specific totally isotropic ideals, the authors establish that every such algebra can be constructed recursively from a trivial or quadratic commutative associative algebra via a finite sequence of these extensions.

Key Contributions and Results

1. Anti-left-invariant Pseudo-Euclidean Nearly Associative Algebras

Nilpotency and Admissibility: The authors prove that any anti-left-invariant pseudo-Euclidean nearly associative algebra is nilpotent of index at most five. Furthermore, they establish that the associated commutative algebra ( $A^+$ ) is a Jacobi-Jordan algebra (also known as a Mock-Lie algebra).
Cyclic Structure: It is shown that the associated pseudo-Euclidean Jacobi-Jordan algebra is cyclic.
Signature Constraints:
- Every anti-left-invariant Euclidean or Lorentzian nearly associative algebra is trivial (i.e., the product is zero).
- Every algebra of signature $(2, n-2)$ is nilpotent of index at most three and is associative.
Classification: Using the double extension method, the paper provides a complete classification of anti-left-invariant pseudo-Euclidean nearly associative algebras of dimension at most six with signature $(2, n-2)$ . The classification relies on the fact that the base algebra in the extension chain must be trivial.

2. Left-invariant Pseudo-Euclidean Nearly Associative Algebras

Solvability and Cyclicity: The associated Lie algebra ( $A^-$ ) is proven to be solvable of index at most two. Additionally, the associated pseudo-Euclidean Lie algebra is cyclic.
Euclidean Case: Every left-invariant Euclidean nearly associative algebra is shown to be a quadratic commutative associative algebra.
Recursive Construction: The authors introduce block, planar, and linear double extensions. They prove that every left-invariant pseudo-Euclidean nearly associative algebra can be recursively constructed from a quadratic commutative associative algebra.
- Over the field of real numbers, this construction can be achieved specifically using planar double extensions.
Classification: A complete classification is provided for left-invariant Lorentzian non-commutative nearly associative algebras of dimension at most four. The results indicate that no such algebras exist in dimension two, and specific structures are identified for dimensions three and four.

Significance
The paper claims to provide a unified framework for describing and classifying pseudo-Euclidean nearly associative algebras in both anti-left-invariant and left-invariant settings. By establishing a correspondence between these algebras and the Levi-Civita products of their associated Lie and Jordan structures, the work bridges differential geometry and non-associative algebra. The introduction of the double extension procedure serves as a powerful tool for generating and classifying these structures, demonstrating that they are not isolated examples but can be systematically built from simpler, well-understood quadratic commutative associative algebras. The results refine the understanding of the geometric properties (flatness, curvature) inherent in these algebraic structures by translating them into precise algebraic identities and structural theorems.

The Great Algebraic Construction Project

The Two Types of Dance Floors

The Magic of Double Extension

Classifying the Small Ones

What This Means

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