Classification of four-dimensional anti-dendriform algebras whose associated associative algebra has the center of dimension one

Artem Lopatin; Jobir Adashev; Shokhsanam Solijonova; Zafar Normatov

REVIEW 3 major objections 4 minor 34 references

Classification of four-dimensional anti-dendriform algebras whose associated associative algebra has the center of dimension one

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Every four-dimensional complex anti-dendriform algebra whose associated associative algebra has one-dimensional center is isomorphic to exactly one of the 47 algebras AD1_4 through AD47_4.

desk verdict Theorem 3.8's parameter α is not an isomorphism invariant—the swap e1↔e2 identifies AD47_4(α) with AD47_4(−α)—so the classification as stated overcounts and needs a sign quotient. read the letter →

arxiv 2412.05295 v1 pith:VXZNO67S submitted 2024-11-24 math.RA

Jobir Adashev , Artem Lopatin , Zafar Normatov , Shokhsanam Solijonova This is my paper · ORCID

classification math.RA MSC 16P1017A30

keywords anti-dendriformalgebraclassificationone-dimensionalcenternilpotentassociativefour-dimensionalcomplexalgebrasisomorphismclassesquotientbyadmissible

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

An anti-dendriform algebra is a vector space with two bilinear operations whose sum is associative. This paper proves that every four-dimensional complex anti-dendriform algebra whose associated associative algebra has a one-dimensional center is isomorphic to exactly one of the 47 algebras AD1_4 through AD47_4 listed in the paper. The proof divides the problem by the center: in each compatible structure the one-dimensional center of the associative algebra is shown to be the full anti-dendriform center, so the quotient is a three-dimensional anti-dendriform algebra taken from an already known classification. The authors work through the eight four-dimensional associative algebras with one-dimensional center that can support such a structure, while the null-filiform algebra As16_4 cannot. The result is a complete explicit normal-form list for this entire family.

What carries the argument

The reduction machinery is the quotient by the common center: Proposition 2.3 says that when the center of the associated associative algebra equals the center of the anti-dendriform algebra, the quotient inherits a compatible anti-dendriform structure of one dimension lower. The paper pairs this with Theorem 2.7, the known classification of all three-dimensional complex anti-dendriform algebras, and Theorem 2.8, the known list of four-dimensional nilpotent indecomposable associative algebras. In each theorem, the authors first use identity (2.6) to force all products involving the central element e4 into one direction, typically giving e4 ⊲ ei = 0 and ei ⊳ e4 = 0, verify that ⟨e4⟩ is the full center, then enumerate the possible three-dimensional quotients and normalize the lifted structure constants by the automorphism group of the associative algebra.

What would settle it

Choose one of the eight associative algebras, write down the most general pair of bilinear operations ⊲ and ⊳ whose sum is the associative product, and impose identities (2.2)–(2.8). If any solution has an anti-dendriform center strictly larger than ⟨e4⟩, or is not isomorphic to one of the listed AD1_4 through AD47_4 tables, the classification is incomplete; conversely, verifying the seven identities and the exact center condition on each of the 47 listed algebras would confirm the exhaustive enumeration.

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Extended reading notes

Core claim

The central claim is stated as Theorems 3.1 through 3.8: up to isomorphism, every four-dimensional complex anti-dendriform algebra associated to a four-dimensional associative algebra with one-dimensional center is one of the pairwise non-isomorphic algebras AD1_4, ..., AD47_4. The only associative algebras that have to be considered are As3_4, As6_4, As8_4, As9_4, As10_4, As13_4, As14_4, and As15_4(α), because Theorem 2.8 lists them as the four-dimensional nilpotent indecomposable associative algebras with one-dimensional center and a previous result excludes a compatible structure on As16_4. For each of these algebras, the paper lifts every possible three-dimensional quotient structure, imposes the seven defining identities (2.2)–(2.8), kills the remaining parameters with the automorphism group of the associative algebra, and obtains a finite list with continuous parameters in some families.

Load-bearing premise

The enumeration is complete only if every compatible anti-dendriform structure on these associative algebras has exactly the same one-dimensional center as the associative algebra itself; a structure with any extra central element would survive the quotient step but would not be represented as a three-dimensional anti-dendriform algebra from the known list.

Editorial extensions

If this is right

Every four-dimensional complex anti-dendriform algebra of this type has a canonical representative in the table AD1_4 through AD47_4, so questions about such algebras can be answered by checking the table.
No compatible anti-dendriform structure exists on the null-filiform algebra As16_4, so the eight algebras considered in Theorems 3.1–3.8 are exactly the possible underlying associative algebras in this family.
The explicit multiplication tables give concrete models for computing degenerations, deformations, and cohomology of anti-dendriform algebras in dimension four.
Any future full classification of four-dimensional anti-dendriform algebras must contain this list as the one-dimensional-center stratum, with the remaining work lying in the cases where the associative center has dimension two or more.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An extension of this approach to dimension five would need either a general center-coincidence lemma for nilpotent associative algebras or a separate treatment of structures whose anti-dendriform center is larger than the associative center.
The parameter families, such as AD5_4(α), AD9_4(α,β), and AD13_4[λ](α,β,γ), are natural candidates for stratification in a geometric classification, since continuous parameters usually correspond to components and degeneration arrows.
A computer algebra re-check of the 47 tables against the identities, together with an invariant-based isomorphism test using center dimension, derived series, and annihilator dimension, would be a cheap independent check of the enumeration.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The short version: the paper's 4-dimensional list is new and the quotient-by-center strategy is sound, but Theorem 3.8 is not a pairwise non-isomorphic classification as written. The stress-test note is correct. Swapping e1 and e2 gives an isomorphism AD47_4(α) ≅ AD47_4(−α), and the same ambiguity is inherited by AD43_4 through AD46_4 and by the underlying associative family As15_4(α). So the enumeration is an overcount unless α is taken modulo sign. That is a load-bearing flaw in the central claim, not a typo.

What the paper does well: it carries out a large and systematic structure-constant computation, deriving constraints from the anti-dendriform identities, using automorphism groups to normalize parameters, and eliminating many cases by contradiction. The 4-dimensional result genuinely goes beyond the earlier 3-dimensional classification and fills a natural row in the program. The quotient-by-center framework is applied carefully in most cases, with the relevant vanishing products checked before asserting that the associative center coincides with the anti-dendriform center.

Soft spots, in order of importance. First, the overcount in Theorem 3.8; this is the main issue and it is fixable by replacing α with a fundamental domain under α∼−α. Second, several isomorphism claims are compressed as “it is not difficult to show”; a referee should ask the authors to spell out the key parameter normalizations, especially in Theorems 3.1 and 3.2. Third, the condition “γ ≥ 0” in Theorem 3.1 is undefined for a complex parameter; the authors likely mean a real nonnegative scalar and should say so. Fourth, there are small internal typos, e.g., “AD51_4” in the proof of Theorem 3.6 should be “AD32_4.” None of these undermine the completeness of the enumeration, only its exactness as stated.

Who this is for: anyone working on algebraic classification of non-associative algebras, especially anti-dendriform algebras and their geometric classification. The paper deserves a serious referee, and I would send it out, but acceptance should require the α-modulo-sign fix and a few clarifications. As it stands I would not cite it without a caveat.

Referee Report

3 major / 4 minor

Summary. The paper classifies four-dimensional complex anti-dendriform algebras whose associated associative algebra is nilpotent with one-dimensional center. The method takes the quotient by the associative center, applies the authors' earlier classification of three-dimensional anti-dendriform algebras (Theorem 2.7), and then solves the structure-constant equations for each possible lift. The main results are Theorems 3.1 through 3.8, which list 47 algebras AD1_4 through AD47_4 and claim that every such algebra is isomorphic to exactly one of them.

Significance. If correct, the paper provides a complete classification of a nontrivial class of low-dimensional anti-dendriform algebras and demonstrates a transfer of a classification problem to a previously solved case through a center quotient. The explicit families and the systematic use of the three-dimensional classification are useful for further studies of anti-dendriform algebras. The main weakness is that the pairwise non-isomorphism part of the classification is not fully established in the text, and there is a concrete overcount in Theorem 3.8.

major comments (3)

[Theorem 3.8, AD47_4(α)] The parameter α in AD47_4(α) is not an isomorphism invariant. The linear map φ(e1)=e2, φ(e2)=e1, φ(e3)=e3, φ(e4)=e4 is an isomorphism of anti-dendriform algebras AD47_4(α) ≅ AD47_4(−α), since φ(e1)⊲φ(e2)=e2⊲e1=−αe4=(−α)φ(e4) and all other products match by the same swap. Hence the family AD47_4(α) with α∈C is not pairwise non-isomorphic, and the exactness of the enumeration in Theorem 3.8 fails as stated. The same basis swap shows As15_4(α) ≅ As15_4(−α) in Theorem 2.8, so the parameter α in that classification also carries a redundancy. This is not a typographical issue but a load-bearing error in the central claim that each algebra is isomorphic to exactly one listed representative.
[Theorems 3.1, 3.2, 3.5, 3.8 (pairwise non-isomorphism)] The proofs repeatedly assert "it is not difficult to show that the constructed algebras are isomorphic" (for example, on page 7 in the proof of Theorem 3.1, and analogous phrases in Theorems 3.2, 3.5, and 3.8) without supplying the actual automorphism computations or isomorphism invariants. Since the theorem statements assert that the listed algebras are pairwise non-isomorphic and that every algebra is isomorphic to exactly one of them, the non-isomorphism proofs are an essential part of the classification. They should be written out or replaced by a certified computation.
[Proofs of Theorems 3.1, 3.2, 3.5, 3.8 (completeness of the 2-nilpotent cases)] The transition from the solved structure constants to the listed algebras is compressed. For instance, in the AD3_3 case of Theorem 3.1, the text states "According to Theorem 2.8 there are five non-isomorphic four-dimensional indecomposable associative 2-nilpotent and the three generated algebras. Hence, we get AD1_4 − AD5_4(α)" without showing how the parameters α11, α12, α21, α22 are reduced and how the ⊳ products are constrained. A complete proof should present the parameter reductions or provide a verifiable computational appendix, because the claim that the list is exhaustive depends on this step.

minor comments (4)

[Proof of Theorem 3.6] On page 23, the text says "we obtain the algebra AD51_4(α,β,γ)"; this should be AD32_4(α,β,γ) to match the theorem statement.
[Theorem 3.1, AD13_4[λ]] The condition "if λ = 0, then γ ≥ 0" is ambiguous over the complex field; please clarify whether γ is meant to be real nonnegative or whether some equivalence relation on γ is intended.
[Proofs of Theorems 3.1–3.8, center claim] The phrase "Then it is easy to see that ⟨e4⟩ is the center" appears in multiple proofs; the derivations only show that e4 lies in the anti-dendriform center. Since the quotient method only requires ⟨e4⟩ to be an ideal, not necessarily the full center, the authors should state this weaker condition explicitly.
[Introduction] The introduction says the paper classifies anti-dendriform algebras "associated with null-filiform associative algebras and three-dimensional algebras"; this appears to be a typo, as the abstract and theorems concern four-dimensional algebras whose associated associative algebra has one-dimensional center.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the 4D classification is derived by solving new structure-constant equations and does not reduce to its cited inputs.

full rationale

The derivation chain is: fix a 4D nilpotent associative algebra with one-dimensional center (Theorem 2.8, an external classification), invoke the center-quotient lemma (Proposition 2.3) and the 3D anti-dendriform classification (Theorem 2.7), then solve the anti-dendriform identities (2.2)-(2.8) for the lifted structure constants, normalize via the automorphism groups listed in Theorem 2.8, and record the surviving algebras AD1_4 through AD47_4. The 3D list [2] is a genuine input, not an assumption equivalent to the 4D conclusion; the structure constants of the 4D algebras contain parameters and products (e.g., e3-terms and parameters α, β, γ, δ, λ) that are not present in the 3D list and are determined by new polynomial equations. Proposition 2.3 is a general lemma from the authors' prior work, but it is not the target classification. The occasional 'it is easy to see that ⟨e4⟩ is the center' statements are terse, and the claimed pairwise non-isomorphism in Theorem 3.8 is a correctness risk (AD47_4(α) and AD47_4(−α) appear isomorphic by swapping e1 and e2), but neither constitutes a circular reduction of the conclusion to the assumptions. No fitted parameter is relabeled as a prediction, and no uniqueness claim is imported from the authors' prior work to forbid alternatives. The central classification therefore has independent computational content.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The classification rests on standard nilpotence facts and on two prior classifications imported from the literature, one of which is the authors' own three-dimensional paper. No new entities or fitted constants are introduced; the parameters in the output families are classification moduli, not fitted data.

assumptions (5)

standard math Every finite-dimensional associative algebra without a non-zero idempotent is nilpotent.
Invoked in Corollary 2.5 following Proposition 2.4 to restrict the associated associative algebra to the nilpotent case.
domain assumption Theorem 2.8: the classification of four-dimensional complex nilpotent indecomposable associative algebras is complete and correct.
Taken from [5] and [30], this list is the starting set of possible associated algebras; if it has gaps or errors, the 4-dimensional classification inherits them.
domain assumption Theorem 2.7: the three-dimensional anti-dendriform classification from [2] is complete and pairwise non-isomorphic.
The quotient As/⟨e4⟩ is assumed to be one of AD3_3 through AD23_3; the 4-dimensional classification is organized as lifts of these algebras.
domain assumption Proposition 2.3: quotient of a compatible anti-dendriform algebra by its center preserves the anti-dendriform structure.
Used to justify that each 4-dimensional algebra induces a 3-dimensional anti-dendriform quotient; cited from [2].
ad hoc to paper For each considered associative algebra, the center ⟨e4⟩ of the associative algebra is also the anti-dendriform center of any compatible structure.
The proofs derive some vanishing products and then state "it is easy to see" that ⟨e4⟩ is the center; this is a non-trivial hypothesis required by the quotient method.

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Pith. "Pith review of Classification of four-dimensional anti-dendriform algebras whose associated associative algebra has the center of dimension one." pith.science (2026). https://pith.science/paper/VXZNO67S

@misc{pith2026241205295,
  author       = {Pith},
  title        = {Pith review of: Classification of four-dimensional anti-dendriform algebras whose associated associative algebra has the center of dimension one},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VXZNO67S}},
  note         = {Machine review of arXiv:2412.05295}
}

read the original abstract

This article is devoted to the classification of anti-dendriform algebras that are associated with associativity. They are characterized as algebras with two operations whose sum is associative. In the paper all four-dimensional complex anti-dendriform algebras associated to four-dimensional associative algebras with one-dimensional center are classified

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

3, 441–483

Abdurasulov K., Adashev J., Eshmeteva S., Transposed Poisson structures on solvable Lie algebras wit h ﬁliform nilradical, Communications in Mathematics, 32 (2024), no. 3, 441–483

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Abdurasulov K., Adashev J., Normatov Z., Solijonova Sh. , Classiﬁcation of three dimensional anti-dendriform algebras, Communications in Algebra, DOI:10.1080/00927872.2024.2 426037

work page arXiv 2024

[3] [3]

Alvarez M.A., Kaygorodov I., The algebraic and geometric classiﬁcation of nilpotent wea kly associative and sym- metric Leibniz algebras, Journal of Algebra, 588 (2021), 278-314

work page 2021

[4] [4]

11, 1071 06

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work page 2022

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Basri W., Rakhimov I.S. and Rikhsiboev I.M., Four-dimensional nilpotent diassociative algebras, J. Generalized Lie Theory Appl., 9 (2015), no. 1, art. ID 1000218, 7 pp

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