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Right representations of Novikov algebras

This paper introduces the concept of right representations for Novikov algebras, characterizing irreducible representations via maximal modular right ideals, establishing a hereditary radical for algebras lacking such representations, defining primitive algebras through almost faithful irreducible representations, and proving that the Jacobson radical equals the intersection of the quasi-kernels of all irreducible representations.

Original authors: A. S. Panasenko

Published 2026-06-16

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CC BY 4.0

Original authors: A. S. Panasenko

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 a universe of mathematical structures called Novikov algebras. You can think of these as a special kind of "rulebook" for how numbers or objects interact when you multiply them. Unlike the standard math you learned in school, where the order of operations sometimes doesn't matter (like $2 \times 3 = 3 \times 2$ ), these algebras have very specific, quirky rules about how things combine.

This paper is like a detective story where the author, A. S. Panasenko, tries to understand the "personality" of these algebras by looking at how they behave when they interact with other things. Here is the breakdown of the investigation using everyday analogies:

1. The Concept of "Right Representations" (The Shadow Play)

In math, to understand a complex object, we often look at how it "casts a shadow" on a simpler stage.

The Analogy: Imagine a complex 3D sculpture (the Novikov algebra). To understand it, we shine a light on it and look at its shadow on a 2D wall (a vector space).
The Paper's Claim: The author focuses specifically on right representations. Think of this as a specific angle of light. The paper defines exactly what rules this shadow must follow to be a valid "shadow" of a Novikov algebra. It turns out that for these specific algebras, the "right" shadow behaves differently than a "left" shadow would, so the author decides to only study the right side to avoid confusion.

2. Irreducible Representations (The Atomic Building Blocks)

Once we have these shadows, the author asks: "What are the simplest, unbreakable shadows?"

The Analogy: Think of a complex machine. You can take it apart, but eventually, you reach the smallest gears that cannot be broken down further. These are irreducible representations.
The Discovery: The paper proves that every one of these "atomic" shadows is essentially a quotient (a slice) of the original algebra.
- If the shadow is "associative" (behaves like normal multiplication), it's a slice of the algebra that acts like a simple field (like a single number system).
- If the shadow is "non-associative" (weird), it's a slice cut by a maximal right ideal.
- The "Modular" Twist: The author introduces a special type of slice called a modular ideal. Think of this as a slice that has a "handle" or a "key" (an element $e$ ) that lets you reconstruct the whole shape from the slice. Without this handle, the slice is useless for understanding the whole.

3. The Jacobson Radical (The "Bad" Part)

Every algebra has a part that is "messy" or "degenerate"—a part that doesn't produce any interesting shadows.

The Analogy: Imagine a fruit basket. Some fruits are rotten. The Jacobson Radical is the collection of all the rotten fruit. If you throw away the radical, you are left with the fresh, good fruit (the "semisimple" part).
The Paper's Claim: The author proves that the class of algebras that have no irreducible shadows at all forms a "radical."
The Big Result: The author defines a quasi-kernel. This is the largest "safe zone" inside the shadow's blind spot. The paper proves that the Jacobson Radical of the entire algebra is exactly the intersection (the common overlap) of all these quasi-kernels from every possible irreducible shadow.
- In plain English: If you take every possible "atomic shadow" the algebra can cast, find the part of the algebra that is invisible in all of them, and combine those invisible parts, you get the Jacobson Radical.

4. Primitive Algebras (The "Pure" Ones)

The author also defines primitive algebras.

The Analogy: A primitive algebra is like a pure, high-quality crystal. It has a "shadow" that is almost perfectly clear (an "almost faithful" representation), meaning the shadow reveals almost everything about the crystal without hiding any major flaws.
The Connection: The paper proves that an algebra is "primitive" if and only if it has such a clear shadow. Furthermore, the Jacobson Radical is simply the intersection of all these "pure" (primitive) parts.

5. The Hereditary Property (Family Traits)

Finally, the paper asks: "If I take a piece of a Novikov algebra (an ideal), does it keep the same 'badness' (radical) as the whole?"

The Analogy: If a family has a genetic trait (like a specific eye color), does a child inherit it?
The Conclusion: Yes. The paper proves the Jacobson radical is hereditary. This means the "rotten fruit" inside a small piece of the algebra is exactly the same as the "rotten fruit" inside the big basket, restricted to that piece. You don't get new rot just by cutting the fruit; you just find the rot that was already there.

Summary

In simple terms, this paper builds a toolkit to dissect Novikov algebras:

It defines how to look at them through "right shadows."
It identifies the "atomic" shadows (irreducible representations).
It proves that the "bad" part of the algebra (the Jacobson Radical) is exactly the part that remains invisible in all of these atomic shadows.
It confirms that this "badness" behaves consistently, whether you look at the whole algebra or just a small piece of it.

The author does not claim these algebras are used for medical cures or engineering; the work is purely about understanding the internal logic and structure of these mathematical objects.

Technical Summary: Right Representations of Novikov Algebras

Problem Statement
The paper addresses a gap in the structural theory of Novikov algebras: the lack of a developed theory regarding right modules and right representations. While the theory of finite-dimensional Novikov algebras (including classifications of simple and semisimple algebras over fields of characteristic 0 and $p>0$ ) and the study of bimodules over simple algebras have been established by authors such as Zel'manov, Osborn, and Xu, the specific concepts of right representations and right modules for arbitrary Novikov algebras had not been previously investigated. The author aims to establish the foundations of right representation theory, define irreducible representations, and characterize the Jacobson radical within this context.

Methodology
The study is conducted over a field $F$ of characteristic different from 2. The methodology relies on the following structural approaches:

Definition of Right Representations: Following the framework of Slinko and Shestakov for general varieties of algebras, the author defines a right representation $\rho: A \to \text{End}(V)$ via $R$ -identities (identities of the right multiplication operator). This leads to the definition of a right $A$ -module $M$ satisfying specific identities derived from the Novikov axioms: $(ma)b = (mb)a$ and $(m, a, b)c = (mc, a, b) = (m, ac, b)$.
Classification of Irreducible Modules: The paper analyzes irreducible modules by distinguishing between associative and non-associative cases. It utilizes the concept of the quasi-kernel $K_\rho(A)$ , defined as the largest ideal of $A$ contained in the kernel of the representation $\rho$ .
Modular Ideals: The author introduces the concept of a right modular ideal. A right ideal $I \neq A$ $I \neq = A$ is modular if either:
- There exist $e, f \in A$ such that $a - (a, e, f) \in I$ for all $a \in A$ (the non-associative case); or
- $I$ is a two-sided ideal such that $A/I$ is a commutative algebra with a unit (the associative case).
Radical Theory: The paper employs the Kurosh-Amitsur radical theory framework to define a radical class $\mathcal{J}$ consisting of Novikov algebras that possess no irreducible modules.

Key Contributions and Results

Description of Irreducible Representations:
The paper proves that every irreducible right representation of a Novikov algebra $A$ is isomorphic to a quotient module $A/I$ , where $I$ is a maximal right modular ideal. Specifically:
- If the module is associative, $I$ is a maximal two-sided ideal, and $A/I$ is a field.
- If the module is non-associative, $I$ is a maximal right ideal, and there exist elements $e, f \in A$ such that $a - (a, e, f) \in I$ for all $a \in A$ .
- The paper provides an example (based on Osborn's work) of a finite-dimensional simple Novikov algebra possessing a non-associative irreducible module, demonstrating that the structure differs significantly from associative or alternative algebras.
The Jacobson Radical:
The author defines the Jacobson radical $J(A)$ of a Novikov algebra as the class $\mathcal{J}$ of algebras with no irreducible representations.
- Theorem: The class $\mathcal{J}$ is a radical class (satisfying conditions of homomorphic images, extensions, and unions of chains).
- Characterization: The Jacobson radical $J(A)$ is proven to be the intersection of the quasi-kernels of all irreducible representations of $A$ .
- Distinction from Classical Cases: The paper notes that, unlike in associative algebras, the intersection of the kernels (not quasi-kernels) of irreducible representations need not be an ideal, nor does it necessarily coincide with the radical.
Primitive Algebras:
A Novikov algebra is defined as primitive if it possesses a maximal modular right ideal that contains no non-zero two-sided ideals.
- Equivalence: A Novikov algebra is primitive if and only if it possesses an almost faithful irreducible representation (where the quasi-kernel is zero).
- Structure: Every primitive Novikov algebra is prime.
Relationship between Radical and Primitive Ideals:
The paper establishes that the Jacobson radical is the intersection of all primitive ideals of the algebra. Consequently, every $J$ -semisimple Novikov algebra is a subdirect product of primitive algebras.
Hereditary Property:
The Jacobson radical of Novikov algebras is proven to be hereditary. This means that for any Novikov algebra $A$ and any ideal $I \subseteq A$ , the radical of the ideal coincides with the intersection of the radical of the algebra and the ideal: $J(I) = J(A) \cap I$ .
Finite-Dimensional Case:
For finite-dimensional Novikov algebras, the Jacobson radical coincides with the largest solvable ideal.

Significance and Claims
The paper claims to provide the first systematic study of right representations for Novikov algebras. Its primary significance lies in:

Extending the structural theory of Novikov algebras to include module theory, specifically addressing the non-associative nature of these algebras which necessitates a distinct definition of modular ideals compared to associative algebras.
Establishing a robust definition of the Jacobson radical for Novikov algebras via the intersection of quasi-kernels of irreducible representations, rather than simple kernels.
Proving that the class of algebras without irreducible representations forms a hereditary radical, thereby integrating Novikov algebras into the broader framework of Kurosh-Amitsur radical theory.
Clarifying the relationship between primitivity and the existence of almost faithful irreducible representations, analogous to classical results in associative algebra theory but adapted for the Novikov variety.

The author explicitly avoids claiming that these results immediately resolve all structural questions or that they apply to left representations (which may differ due to the lack of anti-isomorphism closure in the Novikov variety). The work is presented as a foundational step in the representation theory of Novikov algebras.