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Hom-unitality and hom-associative structures

This paper establishes new characterizations of multiplicative hom-associative structures on one-sided unital algebras by linking them to idempotents and multiplication operators, while also employing a unitalization process to describe hom-unities in non-unital cases and applying these insights to various non-associative algebra classes like Cayley-Dickson and Leibniz algebras.

Original authors: Germán García Butenegro, Abdennour Kitouni, Sergei Silvestrov

Published 2026-03-27

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CC BY 4.0

Original authors: Germán García Butenegro, Abdennour Kitouni, Sergei Silvestrov

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 master architect who loves building structures (algebras) where the order of operations matters. In a perfect, classical building, if you stack bricks A, B, and C, it doesn't matter if you glue A to B first, then C, or glue B to C first, then A. The result is the same. This is what mathematicians call associativity.

But what if the world is a bit "glitchy"? What if the rules of construction change slightly depending on when you look at the bricks? Maybe the glue behaves differently if you apply it on a Tuesday versus a Wednesday. This is the world of Hom-algebras.

This paper is a deep dive into a specific type of "glitchy" building called a Hom-associative algebra. The authors, García Butenegro, Kitouni, and Silvestrov, are trying to figure out how these structures behave when they have a "Master Key" (a unity element) that unlocks the rules.

Here is the breakdown of their discovery, using everyday analogies:

1. The "Glitch" (The Twisting Map)

In a normal algebra, the rule is: $(A \times B) \times C = A \times (B \times C)$ .
In a Hom-associative algebra, the rule is twisted:
$\alpha(A) \times (B \times C) = (A \times B) \times \alpha(C)$ .

Think of $\alpha$ (the twisting map) as a filter or a lens. When you look at the bricks through this lens, the rules of how they stick together change. The paper asks: What happens if our building has a "Master Key" (a unity) that works even through this lens?

2. The "Master Key" (Hom-Unities)

In a normal building, the Master Key (let's call it $1$) is special because $1 \times \text{Anything} = \text{Anything}$ .
In this glitchy world, the "Hom-Key" ( $\omega$ ) doesn't just give you the thing back; it gives you the thing through the lens.
So, $\omega \times \text{Anything} = \alpha(\text{Anything})$ .

The authors realized something brilliant: The lens ( $\alpha$ ) is actually just a multiplication by a specific brick.
If you have a Master Key, the "glitch" isn't a mysterious external force; it's just the building multiplying everything by a specific, hidden brick. This is like realizing that a "magic spell" is actually just a specific type of glue you apply to every brick.

3. The "Perfect" Glue (Multiplicativity)

Sometimes, the lens is "honest." If you look at two bricks stuck together through the lens, it's the same as looking at them separately and then sticking them together. Mathematicians call this multiplicative.
The paper proves a surprising fact: For a building to have this "honest" lens, the Master Key must be an "Idempotent."

Analogy: An idempotent is like a "self-repeating" button. If you press it once, it does something. If you press it again, it does the exact same thing. It doesn't change the state further.
The Discovery: If the Master Key is an idempotent (a "self-repeating" brick), then the whole structure behaves very predictably. If it's not an idempotent, the structure is chaotic and can't be "unital" (it can't have a proper Master Key).

4. The "One-Sided" vs. "Two-Sided" Doors

The paper distinguishes between buildings with a Master Key that works from the Left (you can only enter from the left door) and those with a Two-Sided key (you can enter from both).

Two-Sided: The rules are very strict. The "glitch" (the lens) must be made of a "central" brick—one that plays nice with everyone.
One-Sided: The rules are looser. You can have a Master Key that works from the left but not the right. The authors found that in these looser buildings, the "glitch" is a mix of a "good" brick and a "zero" brick (a brick that dissolves everything it touches).

5. The "Ghost" Buildings (Non-Unital Algebras)

What if the building has no Master Key at all?
The authors use a clever trick called Unitalization. Imagine you have a broken house with no front door. You can't fix the house, so you build a new house around it that does have a front door, and the old house is just a room inside the new one.
By studying this "new house," they can figure out the rules for the "old house." They found that even in buildings without keys, there are "ghost keys" hidden inside the structure (specifically in the "nilpotent" parts, which are like bricks that eventually crumble to dust if you hit them enough times).

6. The Big Takeaway

The paper connects three seemingly different concepts:

The Lens ( $\alpha$ ): How the rules are twisted.
The Master Key ( $\omega$ ): The element that defines the twist.
Idempotents: The "self-repeating" bricks that make the system stable.

The Conclusion:
If you want to build a stable, "glitchy" structure with a Master Key, you are essentially looking for a specific type of brick (an idempotent) that acts as the lens. If you find one, the whole structure becomes much easier to understand. If you don't, the structure is either too chaotic to have a key, or it collapses into a simpler, classical structure.

Why does this matter?
These structures aren't just abstract math games. They appear in Quantum Physics and String Theory, where the rules of space and time might be "twisted" or "discrete" (like pixels on a screen) rather than smooth. Understanding how these "twisted" rules work with "keys" helps physicists model the universe when things get weird at the smallest scales.

In a nutshell: The authors found that in a world where math rules are slightly distorted, the "keys" to the system are actually just specific, self-repeating numbers. If you find the right self-repeating number, you can unlock the secrets of the distortion.

1. Problem Statement

The paper addresses the structural characterization of hom-associative algebras, specifically focusing on the interplay between unitality (the existence of a unity element) and hom-associativity (associativity deformed by a linear twisting map $\alpha$ ).

While classical unitality implies strict associativity under certain conditions in hom-algebras, the authors investigate hom-unitality, a generalized concept where a "hom-unity" $\omega$ satisfies $\omega \cdot x = \alpha(x)$ (left) or $x \cdot \omega = \alpha(x)$ (right). The core problems addressed are:

How does the existence of one-sided or two-sided unities constrain the twisting map $\alpha$ ?
Under what conditions is a hom-associative algebra multiplicative (i.e., $\alpha(xy) = \alpha(x)\alpha(y)$ )?
How can the subspaces of elements inducing hom-associative structures (hom-unities) be explicitly characterized in one-sided unital, two-sided unital, and non-unital algebras?
How do these structures behave in specific non-associative classes like Cayley-Dickson algebras, Leibniz algebras, and hom-Leibniz algebras?

2. Methodology

The authors employ a rigorous algebraic approach combining:

Operator Theory: Representing the twisting map $\alpha$ as multiplication operators ( $L_a$ or $R_a$ ) derived from unity elements.
Subspace Decomposition: Defining and analyzing specific subspaces of the algebra (e.g., centers, nuclei, annihilators, and eigenspaces) to characterize hom-unities.
Unitalization: For non-unital algebras, the authors utilize a standard unitalization process (embedding the algebra $A$ into $\tilde{A} = A \oplus \mathbb{F}$ ) to extend results from unital cases to non-unital ones.
Case Studies: Applying general theorems to concrete algebraic structures, including Cayley-Dickson algebras (Complex numbers, Quaternions, Octonions, Sedenions) and Leibniz algebras.

3. Key Contributions and Results

A. One-Sided Unital Hom-Associative Algebras

Twisting Map Characterization: The authors prove that in any one-sided unital hom-associative algebra, the twisting map $\alpha$ is necessarily a multiplication operator by the element $\alpha(1_\star)$ . Specifically, $\alpha = L_{\alpha(1_l)}$ for left-unital algebras.
Multiplicativity vs. Idempotency: A central result (Theorem 16) establishes that for a unital hom-associative algebra, the twisting map $\alpha$ is multiplicative if and only if the hom-unity $\alpha(1_\star)$ is idempotent ( $\alpha(1_\star)^2 = \alpha(1_\star)$ ).
Consequence: If a unital hom-associative algebra has no non-trivial idempotents (e.g., division algebras), it cannot be a non-trivial multiplicative hom-associative algebra unless it is strictly associative.

B. Two-Sided Unital Algebras

Subalgebra of Hom-unities: The paper provides an explicit formula for the subalgebra $AC(A)$ of elements inducing hom-associative structures in two-sided unital algebras:
$AC(A) = Z(A) \cap N(A) \cap \text{Ann}_A([A, A, A]_{as})$
where $Z(A)$ is the center, $N(A)$ is the nucleus, and $[A, A, A]_{as}$ is the associator subspace.
Cayley-Dickson Application: The authors compute $AC(A)$ for the first Cayley-Dickson algebras:
- Complex Numbers ( $\mathbb{C}$ ): $AC(\mathbb{C}) = \mathbb{C}$ (all elements induce hom-associativity).
- Quaternions ( $\mathbb{H}$ ): $AC(\mathbb{H}) = \mathbb{R}$ (only real multiples of unity).
- Octonions ( $\mathbb{O}$ ) & Sedenions ( $\mathbb{S}$ ): $AC(\mathbb{O}) = AC(\mathbb{S}) = \{0\}$ . This implies these algebras admit no non-trivial hom-associative structures induced by multiplication.

C. One-Sided Unital Algebras (General Case)

Decomposition: The space of hom-unities $AC_l(A)$ is decomposed as:
$AC_l(A) = (AC_l(A) \cdot 1_l) \oplus \text{Ann}_A^l(A)$
The subspace $AC_l(A) \cdot 1_l$ forms a commutative associative subalgebra.
Non-Existence in Domains: The paper proves that there are no non-trivial hom-associative one-sided unital division algebras (Theorem 46). If such an algebra exists, it must be either trivial or strictly associative.
Hom-Leibniz Incompatibility: A significant finding (Theorem 58) is that in a hom-Leibniz algebra, a unity on one side is incompatible with the hom-Leibniz identity on the same side unless the twisting map is zero. However, a left unity is compatible with a right hom-Leibniz identity (and vice versa).

D. Non-Unital Algebras and Unitalization

Unitalization Process: For non-unital algebras, the authors describe hom-unities via the unitalization $\tilde{A}$ . They show that the subalgebra of hom-unities in $\tilde{A}$ contains a "hidden" subspace isomorphic to $HUn(A)$ (the hom-unities of the original algebra) as a two-sided ideal.
Eigenspace Characterization: Hom-unities in non-unital algebras are characterized by eigenspace conditions of multiplication operators relative to the associator.
Leibniz Algebras: The study reveals that hom-associative structures in Leibniz algebras are linked to 3-nilpotent elements. Specifically, $HUn(L)$ consists of elements in the intersection of the center and the annihilator of the derived subalgebra.

4. Significance and Impact

Unification of Concepts: The paper bridges the gap between classical unitality and twisted hom-associativity, showing that multiplicativity in hom-algebras is deeply tied to the idempotency of the unity element.
Classification Tool: The explicit characterization of $AC(A)$ provides a powerful tool for classifying hom-associative structures. It explains why certain algebras (like Octonions) cannot support non-trivial hom-associative deformations via multiplication.
Structural Constraints: The results impose strong constraints on the existence of hom-associative structures in domains and division algebras, effectively ruling out non-trivial multiplicative hom-associative structures in these classes.
New Subclasses: The identification of specific subspaces (like $AC_l(A) \cdot 1_l$ ) as commutative associative subalgebras within non-associative settings offers new avenues for constructing and analyzing hom-algebras.
Applications to Physics: By clarifying the structure of hom-Leibniz and hom-Lie algebras (via Yau twists), the work supports the development of models in mathematical physics where non-associative symmetries are deformed.

In summary, this paper provides a comprehensive structural analysis of hom-unital hom-associative algebras, moving from general definitions to specific characterizations in one-sided, two-sided, and non-unital contexts, while applying these findings to major classes of non-associative algebras.