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Action accessible and weakly action representable varieties of algebras

This paper establishes that the varieties of $k$ -nilpotent Lie algebras ( $k \geq 3$ ) and $n$ -solvable Lie algebras ( $n \geq 2$ ) are action accessible but not weakly action representable, providing the first known examples of such non-associative algebras and refining similar results for nilpotent and solvable groups.

Original authors: Xabier García-Martínez, Manuel Mancini

Published 2026-06-16

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CC BY 4.0

Original authors: Xabier García-Martínez, 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 trying to organize a massive library of mathematical structures called "algebras." In this library, some sections are very well-behaved, while others are a bit chaotic. The paper you are asking about is a detective story where the authors investigate two specific rules for organizing this library: Action Representability and Action Accessibility.

Here is the story in plain English, using some creative analogies.

The Setting: The Library of Algebras

Think of an "algebra" as a set of rules for how things interact (like how numbers add or multiply, but more abstract). Mathematicians study "varieties" of these algebras, which are like different genres of books in our library (e.g., "Lie Algebras," "Groups," "Associative Algebras").

To understand how these algebras behave, mathematicians look at Split Extensions.

The Analogy: Imagine you have a base building (Object $B$ ) and you want to attach a new wing to it (Object $X$ ). A "split extension" is a specific, clean way of attaching that wing so you can easily take it off later if needed.
The Goal: Mathematicians want to know: "Can we build a single 'Master Blueprint' (an object called $[X]$ ) that tells us every possible way to attach this wing to any building?"

The Three Levels of Organization

The paper discusses three levels of how well this library is organized, from strongest to weakest:

Action Representable (The Perfect Library):
- The Rule: For every wing ( $X$ ), there is a single, perfect Master Blueprint ( $[X]$ ). If you want to know how to attach the wing to a building, you just look at the blueprint. It's a perfect 1-to-1 match.
- The Reality: This is very rare. The paper notes that only a few types of algebras (like standard Lie algebras) have this perfect organization.
Weakly Action Representable (The Good Library):
- The Rule: We can't find a perfect Master Blueprint, but we can find a "Candidate Blueprint" ( $T$ ). This candidate might list some impossible ways to attach the wing, but it definitely lists all the real, possible ways. It's like a catalog that has a few extra pages of fiction, but no missing chapters.
- The Expectation: Many mathematicians hoped that if a library was "Action Accessible" (see below), it would automatically be "Weakly Action Representable." They thought the "Good Library" rule would always follow the "Accessible" rule.
Action Accessible (The Manageable Library):
- The Rule: The library is organized enough that we can at least talk about how to attach wings, even if we don't have a single blueprint for it. It's "manageable."
- The Expectation: The authors suspected that "Manageable" might imply "Good," but they weren't sure.

The Big Question

The paper tackles two burning questions:

Does "Manageable" always mean "Good"? (If a variety is Action Accessible, is it necessarily Weakly Action Representable?)
Does the "Good" rule pass down to sub-groups? (If the main library is "Good," are all the smaller sections inside it also "Good"?)

The Investigation: Breaking the Rules

The authors, X. García-Martínez and M. Mancini, act like detectives proving that the answer to both questions is NO.

They use a clever trick borrowed from a previous detective (J. R. A. Gray) involving a concept called the Amalgamation Property.

The Analogy: Imagine you have two different blueprints for a room ( $B$ and $B_1$ ) that both share a small, identical hallway ( $S$ ). The "Amalgamation Property" asks: "Can we glue these two rooms together along that hallway to make one big, valid building ( $D$ )?"
The Twist: The authors find specific types of algebras (specifically $k$ -nilpotent Lie algebras and $n$ -solvable Lie algebras) where you can glue them together in the big, general library, but you cannot glue them together if you are restricted to the smaller, specific section of the library.

The Results: The "No" Answers

By finding these specific cases where the "gluing" fails in the sub-sections, the authors prove:

The "Manageable" $\neq$ "Good" Rule:
They found varieties of algebras (like 3-nilpotent Lie algebras) that are "Action Accessible" (manageable) but fail to be "Weakly Action Representable" (good).
- Translation: Just because a section of the library is organized enough to be studied, it doesn't mean it has a "Good" catalog. The "Good" rule is stricter than we thought.
The Sub-Group Failure:
They proved that even if the main library (Lie algebras) is "Weakly Action Representable," its smaller sub-sections (like the 3-nilpotent ones) are not.
- Translation: Being a "Good" library doesn't guarantee that every small room inside it is also "Good." The rules break down when you zoom in.

The Bonus Discovery

The authors also applied their detective work to Groups (a different type of algebra). They refined an old result to show that:

$k$ -nilpotent groups (for $k \ge 3$ ) are not weakly action representable.
2-solvable groups are not weakly action representable.

The Takeaway

Before this paper, mathematicians hoped that "Action Accessibility" was a strong enough condition to guarantee "Weak Action Representability," and that these properties would stay consistent even in smaller sub-groups.

This paper pulls the rug out from under those hopes. It shows that the mathematical world is more complex:

You can have a library that is organized enough to work with, but still lacks a perfect catalog.
You can have a "Good" library where the smaller rooms are actually "Bad" (in terms of this specific rule).

In short: The rules of the big library do not automatically apply to the small rooms, and being "manageable" doesn't mean you have a "perfect catalog."

Technical Summary: Action Accessible and Weakly Action Representable Varieties of Algebras

Problem Statement
This paper investigates the relationship between two categorical properties within the context of varieties of non-associative algebras over a field: action accessibility and weak action representability.

In semi-abelian categories, internal actions on an object $X$ correspond to split extensions. A category is action representable if the functor of actions is naturally isomorphic to a hom-functor $\text{Hom}(-, [X])$ . A weaker notion, weak action representability, requires only a natural monomorphism $\text{Act}(-, X) \hookrightarrow \text{Hom}(-, T)$ for some object $T$ . It is established that action representability implies weak action representability, which in turn implies action accessibility.

The central problem addressed is whether the converse implications hold, specifically:

Does weak action representability imply action accessibility in the context of non-associative algebras? (The paper notes the converse is known to be false in general, but seeks specific counterexamples in algebras).
Is the property of being weakly action representable preserved under taking subvarieties?

Prior to this work, it was an open question whether there existed an action accessible variety of non-associative algebras that failed to be weakly action representable. Furthermore, while J. R. A. Gray had shown that varieties of $n$ -solvable groups ( $n \geq 3$ ) are not weakly action representable, the status of nilpotent groups ( $k \geq 3$ ) and solvable groups ( $n=2$ ) remained unresolved. Similarly, the behavior of these properties in varieties of Lie algebras (specifically nilpotent and solvable subvarieties) was unknown.

Methodology
The authors employ a method based on the amalgamation property (AP) and a specific proposition by J. R. A. Gray (Proposition 1.3). The core strategy involves demonstrating that a specific variety $\mathcal{X}$ is not weakly action representable by constructing a counterexample to the amalgamation property within that variety, while ensuring the amalgamation exists in a larger, action representable category.

The proof relies on the following logical structure (Proposition 1.3):

Identify a variety $\mathcal{X}$ as a Birkhoff subcategory of an action representable category $\mathcal{C}$ (e.g., $\mathcal{C} = \text{Grp}$ or $\mathcal{C} = \text{Lie}$ ).
Construct two monomorphisms $m, m': S \hookrightarrow B$ and $S \hookrightarrow B'$ within $\mathcal{X}$ that cannot be amalgamated in any object of $\mathcal{X}$ .
Show that these monomorphisms can be amalgamated in the larger category $\mathcal{C}$ via monomorphisms $u, u'$ into an object $D$ .
Demonstrate the existence of an object $X$ in $\mathcal{X}$ and a monomorphism $v: D \hookrightarrow [X]$ (where $[X]$ is the representing object in $\mathcal{C}$ ) such that the split extensions induced by the composition $v \circ u$ and $v \circ u'$ remain within the subvariety $\mathcal{X}$ .

If these conditions are met, $\mathcal{X}$ cannot be weakly action representable.

To execute this, the authors utilize:

For Groups: A specific example by B. H. Neumann involving an abelian group $S$ and 2-nilpotent groups $B, B'$ that cannot be amalgamated in any solvable group. The authors refine this by constructing specific representations into $\text{Aut}(X)$ (where $X = \mathbb{Z}_5^3$ ) to prove the resulting semi-direct products are $k$ -nilpotent or 2-solvable.
For Lie Algebras: An adaptation of Neumann's example using Lie algebras. They construct an abelian algebra $S$ and two copies of the 3-dimensional Heisenberg algebra ( $B, B'$ ). They prove that while $B$ and $B'$ can be amalgamated in $\mathfrak{gl}(3, F)$ , the amalgamation forces the resulting algebra to contain a subalgebra isomorphic to $\mathfrak{sl}(2, F)$ , which is simple and non-solvable. This contradiction proves the impossibility of amalgamation within the variety of solvable (or nilpotent) Lie algebras.

Key Contributions and Results

Refinement of Gray's Result on Groups:
The paper proves that the varieties of $k$ -nilpotent groups (for $k \geq 3$ ) and 2-solvable groups are not weakly action representable. This extends Gray's previous result which covered $n$ -solvable groups for $n \geq 3$ .
- Result: Theorem 2.1 and Corollary 2.2.
New Counterexamples in Non-Associative Algebras:
The paper establishes that the varieties of $k$ -nilpotent Lie algebras (for $k \geq 3$ ) and $n$ -solvable Lie algebras (for $n \geq 2$ ) are not weakly action representable.
- Result: Theorem 3.1 and Corollary 3.2.
- Significance: These are the first known examples of action accessible varieties of non-associative algebras that fail to be weakly action representable.
Resolution of Open Questions:
The results provide negative answers to the open questions posed in prior literature:
- The converse of "weakly action representable $\implies$ action accessible" does not hold in the context of varieties of non-associative algebras.
- The property of being weakly action representable is not preserved under taking subvarieties. Specifically, the variety of Lie algebras ( $\text{Lie}$ ) is action representable (and thus weakly action representable), but its subvarieties of $k$ -nilpotent and $n$ -solvable algebras are not.

Significance
The paper claims that these findings fundamentally clarify the hierarchy of categorical properties in algebraic varieties. By providing concrete counterexamples within the well-studied contexts of Lie algebras and groups, the authors demonstrate that the structural constraints required for weak action representability are not inherited by subvarieties, even when the parent variety possesses the strongest form of action representability. This refines the understanding of the "amalgamation property" as a necessary condition for weak action representability in semi-abelian categories. The work does not propose new applications or future experimental proposals but strictly focuses on the theoretical classification of these algebraic varieties.