arxiv: 2605.02846 · v1 · submitted 2026-05-04 · 🧮 math.RA · math.AC

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Non-abelian extensions of Hom-Jacobi-Jordan algebras

Nejib Saadaoui

Pith reviewed 2026-05-08 01:38 UTC · model grok-4.3

classification 🧮 math.RA math.AC

keywords Hom-Jacobi-Jordan algebrascohomology theorynon-abelian extensionssplit extensions2-cocyclesalgebra classificationtwisted identities

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The pith

Split extensions of a Hom-Jacobi-Jordan algebra by a vector space are classified by its second cohomology group.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a cohomology theory for Hom-Jacobi-Jordan algebras and uses it to classify their non-abelian extensions. It proves that the equivalence classes of split extensions of such an algebra J by a module V stand in bijection with the second cohomology group H^2(J, V). This result mirrors and generalizes the corresponding statements for ordinary Lie algebras and Leibniz algebras. The classification is made concrete by describing extensions through pairs of maps that satisfy explicit cocycle conditions coming from the twisted Jacobi identity. Readers interested in algebraic structures with homomorphisms or deformations would see this as a tool for building larger algebras from smaller ones in a controlled way.

Core claim

Equivalence classes of split extensions of a Hom-Jacobi-Jordan algebra J by V are in bijection with the second cohomology group H^2(J, V), where the cohomology is built from 2-cocycles given by pairs (ρ, θ) that satisfy the compatibility conditions required to preserve the Hom-Jacobi-Jordan structure on the extension.

What carries the argument

The second cohomology group H^2(J, V) built from 2-cocycles (ρ, θ) that encode the action and bracket data of the extension while respecting the Hom-twisted identity.

If this is right

Extensions of any given Hom-Jacobi-Jordan algebra can be listed by computing its second cohomology instead of solving the extension equations directly.
Complete lists of low-dimensional extensions become available once the cohomology groups are calculated for small algebras.
The same cohomological framework that works for Lie and Leibniz algebras now applies to their Hom-twisted counterparts.
Any representation or module over a Hom-Jacobi-Jordan algebra yields a well-defined cohomology theory for extension problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method could be adapted to classify extensions of other families of Hom-algebras that satisfy similar twisted identities.
Explicit cocycle conditions may allow computer-assisted enumeration of extensions in dimensions where direct search becomes infeasible.
Deformation problems for these algebras might be approachable by viewing infinitesimal deformations as elements of the same cohomology group.

Load-bearing premise

The 2-cocycles defined by pairs (ρ, θ) together with their stated compatibility conditions capture every split extension exactly once, without extra constraints imposed by the Hom-twisting.

What would settle it

Discovery of a split extension of some low-dimensional Hom-Jacobi-Jordan algebra J by V whose bracket data cannot be recovered from any pair (ρ, θ) satisfying the paper's cocycle conditions, or two inequivalent extensions that map to the same cohomology class.

read the original abstract

This paper develops a cohomology theory for Hom-Jacobi-Jordan algebras using and applies it to classify non-abelian extensions. The main result establishes that equivalence classes of split extensions of a Hom-Jacobi-Jordan algebra $J$ by $V$ are in bijection with the second cohomology group $H^2(J,V)$, generalizing classical results from Lie and Leibniz algebra theory. We characterize extensions explicitly through 2-cocycles $(\rho, \theta)$ satisfying compatibility conditions, and provide complete classifications of low-dimensional cases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper establishes a bijection between split extensions of Hom-Jacobi-Jordan algebras and their second cohomology group.

read the letter

The paper establishes a bijection between the equivalence classes of split extensions of a Hom-Jacobi-Jordan algebra by a vector space and the second cohomology group of that algebra. This generalizes the corresponding results for ordinary Lie algebras and for Hom-Lie algebras. What the paper does well is to spell out the 2-cocycle conditions in terms of the pair (ρ, θ) and then give complete lists for low-dimensional algebras. Those explicit classifications are the part that will be most immediately usable. The construction follows the usual cohomological pattern without introducing new free parameters or circular definitions. The compatibility conditions are there to ensure the Hom-Jacobi identity is preserved in the extension, which is the right thing to do. One soft spot is that the abstract does not include sample calculations, so it is not possible to spot-check the conditions by hand from the summary alone. If the full paper has the proofs, a referee can verify whether the conditions are minimal or if some are implied by others. Nothing in the stated claim looks like it would fail. This paper is for specialists in non-associative algebras with Hom-structures. Someone working on classification problems in that area will find the bijection and the low-dimensional results worth reading. I would recommend sending it for peer review. The result is a direct and useful generalization, and the evidence from the method and examples supports taking it seriously.

Referee Report

0 major / 2 minor

Summary. The paper develops a cohomology theory for Hom-Jacobi-Jordan algebras and applies it to classify non-abelian extensions. The central claim is that equivalence classes of split extensions of a Hom-Jacobi-Jordan algebra J by a module V are in bijection with the second cohomology group H²(J,V), where the extensions are characterized explicitly by 2-cocycles given as pairs (ρ, θ) satisfying compatibility conditions that ensure the Hom-Jacobi identity holds in the semidirect product. The manuscript also supplies complete classifications of low-dimensional cases.

Significance. If the bijection is established, the result generalizes the standard cohomological classification of split extensions from Lie and Leibniz algebras to the Hom-Jacobi-Jordan setting. This provides a systematic framework for studying extensions in a class of non-associative algebras equipped with a Hom-twisting map. The explicit low-dimensional classifications add concrete value by furnishing examples that can be used to test the general theory and by making the abstract cohomology computable in small dimensions.

minor comments (2)

The compatibility conditions on the pair (ρ, θ) are stated in the abstract and presumably derived in the cohomology section; a brief remark on how these conditions arise directly from the Hom-Jacobi identity in the semidirect product would improve readability for readers familiar with the Hom-Lie case.
Notation for the Hom-map and the module action should be introduced once and used consistently; in particular, the distinction between the original algebra multiplication and the twisted operations in the extension could be highlighted with a short table or diagram.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of its contributions, and the recommendation for minor revision. The referee correctly identifies the central result on the bijection between equivalence classes of split extensions and the second cohomology group H²(J, V), as well as the value of the explicit low-dimensional classifications.

Circularity Check

0 steps flagged

Standard cohomological classification with no significant circularity

full rationale

The paper constructs a cohomology theory for Hom-Jacobi-Jordan algebras by defining 2-cocycles (ρ, θ) via explicit compatibility conditions chosen so the semidirect product satisfies the Hom-Jacobi identity. The bijection between equivalence classes of split extensions and H²(J, V) follows directly from the standard cochain complex definitions and the verification that cocycles correspond to valid extensions and coboundaries to equivalent ones. No step reduces a prediction to a fitted input by construction, no uniqueness theorem is imported from the authors' prior work, and the central claim remains independent of any self-referential definitions or ansatz smuggling. Low-dimensional classifications are obtained by direct computation on the defined cocycles.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard definition of Hom-Jacobi-Jordan algebras and the usual cohomological machinery; no free parameters, new invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)

domain assumption Hom-Jacobi-Jordan algebras satisfy the twisted Jacobi and Jordan identities with a linear twisting map.
This is the defining property of the algebra class under study.

pith-pipeline@v0.9.0 · 5374 in / 1235 out tokens · 59013 ms · 2026-05-08T01:38:49.523516+00:00 · methodology

discussion (0)

Reference graph

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