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Axial Algebras: Questions and Conjectures

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Pith's one-line read The paper expands the list of open problems and conjectures in axial algebras beyond a recent survey.

desk verdict This is a curated list of open problems in axial algebras that expands an existing survey, with no new theorems or proofs. read the letter →

arxiv 2606.30048 v1 pith:7LQGKDXN submitted 2026-06-29 math.RA

I. Gorshkov , S. Shpectorov This is my paper · ORCID
classification math.RA
keywords axialalgebrasopenproblemsconjecturesfusionlawsnon-associativeidempotentsautomorphisms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Axial algebras are non-associative algebras generated by idempotents called axes whose adjoint actions satisfy a fusion law. When the fusion law is graded, the axes produce automorphisms that link the algebras to groups. This article complements an earlier survey by significantly expanding the list of interesting open problems suggested by specialists. It also supplies further discussion of related concepts and available results. A reader would care because the problems point to concrete directions for work in non-associative algebra and its ties to group theory.

What carries the argument

Axial algebra: non-associative algebra generated by idempotents (axes) whose adjoint action satisfies a fusion law, often yielding group automorphisms when the law is graded.

What would settle it

A check showing that several listed problems have already been resolved in the literature would falsify the claim that the paper expands the set of open problems.

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Extended reading notes

Core claim

The authors significantly expand the list of interesting open problems in axial algebras and provide further discussion of related concepts and available results, complementing the recent survey.

Load-bearing premise

The listed problems are genuinely open, significant to specialists, and accurately described relative to the prior survey.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

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Summary. The manuscript is a curated collection of open questions and conjectures in the theory of axial algebras. It complements the recent survey [ms] by expanding the list of interesting open problems suggested by specialists and providing further discussion of related concepts and available results. Axial algebras are defined as non-associative algebras generated by idempotents (axes) whose adjoint actions satisfy a fusion law; when graded, these induce automorphisms linking the algebras to groups.

Significance. This work provides a service to the axial algebra community by systematically expanding the catalog of open problems. If the listed items are accurately described as open and of interest to specialists, the manuscript can help direct future research in non-associative algebras and their connections to finite groups. No new theorems, proofs, or computational results are claimed, so the contribution rests on the utility of the curated list rather than on technical derivations.

minor comments (1)
  1. The abstract and introduction reference the survey as [ms] without a full bibliographic entry visible in the provided text; ensure the reference list includes the complete citation for [ms] to allow readers to locate the prior survey.

Simulated Author's Rebuttal

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We thank the referee for their positive evaluation of the manuscript and for recommending acceptance. The report accurately captures the purpose of the work as a curated expansion of open questions to complement the recent survey.

Circularity Check

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No circularity; paper is a curated list of open questions with no derivations

full rationale

The manuscript contains no theorems, proofs, equations, predictions, or fitted parameters. Its sole purpose is to expand a list of open problems complementing the cited survey [ms]. All references to prior work serve only to frame questions and are not load-bearing for any internal claim. No self-citation reduces a result to an unverified input, and the paper is self-contained as a non-technical survey of open items.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

No free parameters, axioms, or invented entities are introduced because the paper contains no derivations or new constructions; it only lists questions about existing concepts.

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Cite this review

Pith. "Pith review of Axial Algebras: Questions and Conjectures." pith.science (2026). https://pith.science/paper/7LQGKDXN

@misc{pith2026260630048,
  author       = {Pith},
  title        = {Pith review of: Axial Algebras: Questions and Conjectures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7LQGKDXN}},
  note         = {Machine review of arXiv:2606.30048}
}
read the original abstract

Axial algebras are non-associative algebras generated by idempotents, called axes, whose adjoint action satisfies a fusion law. When this fusion law is graded, axes naturally lead to automorphisms of the algebra, and so such axial algebras are inextricably linked with groups. This article is meant to complement the recent survey \cite{ms} by significantly expanding the list of interesting open problems suggested by the specialists in the field, and providing a further discussion of the related concepts and available results.

Figures

Figures reproduced from arXiv: 2606.30048 by the authors.

Figure 1
Figure 1. Monster fusion law This table represents what is now called a fusion law, specifically, the Monster fusion law, and it describes the properties of the adjoint action of Majorana axes. Up to our knowledge, this was the first time a fusion law appeared as part of the axiomatics of a class of algebras. However, similar properties of idempotents in algebras have been observed earlier. In 1881, Peirce [P] described how a… view at source ↗
Figure 2
Figure 2. Associative fusion law In 1947, Albert introduced three-term Peirce decompositions in Jordan algebras. Again, these are with respect to an arbitrary idempotent and they can be described by [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Jordan fusion law 2 Background This section introduces the foundational definitions of axial algebras and their subclasses, focusing on algebras of Jordan and Monster type. Definition 2.1 A fusion law is a pair (F, ∗), where F is a set and ∗ : F × F → 2 F is a binary operation on F with values in the set 2 F of all subsets of F. The fusion laws we focus on are defined on a small set F, and we find it useful to organ… view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: shows the fusion law of Jordan type η, denoted by J (η). Here F = J (η) = {1, 0, η}, η ̸= 1, 0, is a subset of a field F. The fusion law J (η) generalises the Jordan fusion law from [PITH_FULL_IMAGE:figures/full_fig_p003_4.png]
Figure 5
Figure 5. Figure 5: Monster type fusion law M(α, β) (b) For λ ∈ F, set Aλ(a) = {b ∈ A|ada(b) = λb} = {b ∈ A|ab = λb}. (c) For Λ ⊆ F, set AΛ(a) = L λ∈Λ Aλ(a). Let F ⊆ F be a fusion law. Definition 2.3 An idempotent 0 ̸= a ∈ A is an F-axis if (a) A = AF (a); and (b) Aλ(a)Aµ(a) ⊆ Aλ∗µ(a) for…
Figure 6
Figure 6. Figure 6: Fusion law M(4η, 4η) This product also works in the group 52 : 2, but not in A5 if for products of order 1, 2, and 3 the usual Matsuo multiplication is used. 6.2 Almost Jordan type fusion law Whybrow [W] did an algorithmic study of all fusion laws of small size. In par…
Figure 7
Figure 7. Figure 7: Fusion law J (α, β) type. algebras can be classified. This was developed further by Afanasev [Af], who provided a nearly complete classification of 2-generated primitive algebras of almost Jordan type (α, β). More recently, Welch [SW] completed Afanasiev’s classificati…
Figure 8
Figure 8. Figure 8: The fusion law F for Hˆ 5, the eigenvalue 5 2 becomes zero, and the fusion law folds into the Monster type (2, 1 2 ) fusion law. Problem 6.6 Is it possible to classify primitive algebras with the above extended Monster type fusion law. C. Franchi, M. Mainardis 6.4 Axia…
Figure 9
Figure 9. Figure 9: Fusion law H for Hsiang algebras 35 [PITH_FULL_IMAGE:figures/full_fig_p035_9.png]
Figure 10
Figure 10. Figure 10: Fusion law B for type Bn. This is a C2-graded fusion law with the partition F+ = {1, 0} and F− = { 1 2 , 1 4 }. Problem 6.12 Is it possible to classify 2-generated primitive axial algebras with the fusion law B? S. Shpectorov If this first step is doable then the much…
Figure 11
Figure 11. Figure 11: Fusion law CG for type Cn, F4. More generally, all Chayet-Garibaldi algebras, except the one arising for the type G2, have primitive idempotents obeying the fusion law in [PITH_FULL_IMAGE:figures/full_fig_p037_11.png]
Figure 12
Figure 12. Figure 12: Fusion law G for type G2. We will formulate the following blanket questions concerning these fusion laws. Problem 6.14 What can we say about algebras with fusion laws CG and G? E.g., can 2-generated primitive algebras with these fusion laws be classified? S. Shpectoro…

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  1. Fusion rules from the Norton inequality

    math.RA 2026-07 accept novelty 6.5 of 10

    Norton inequality forces A0(e) and A1(e) to be subalgebras and A0(e)A1(e)subseteq A1/2(e) for non-degenerate Frobenius forms; a degenerate counter-example is given.

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