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Transposed $\delta$-Poisson algebra structures on null-filiform associative algebras

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Pith's one-line read The paper classifies all transposed δ-Poisson algebra structures on null-filiform associative algebras, showing the parameter δ acts only through the roots 0, 1, 2 of δ^3−3δ^2+2δ, and proves every ordinary δ-Poisson structure on the same…

desk verdict Sound new classifications for δ=0,1,generic with a clean structural observation, but the completeness claim is impaired by an omitted zero-bracket family and reliance on an unverified overlapping preprint for δ=2. read the letter →

arxiv 2507.10554 v1 pith:2RHMYC5G submitted 2025-06-07 math.RA

Nigora Daukeyeva , Maqpal Eraliyeva , Feruza Toshtemirova This is my paper
classification math.RA MSC 17A3017A5017B63
keywords transposedδ-Poissonalgebranull-filiformassociativePoissonLieclassificationofpolynomialsautomorphismgroup
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

The paper aims to settle, for every value of the parameter δ, the question of which Lie brackets can sit on the n-dimensional null-filiform associative algebra $μ_n^{0}$ so that the transposed δ-Poisson identity holds. The central structural fact is the obstruction ($δ^{3}$−$3δ^{2}$+2δ)e_3·[e_1,e_2]=0, which splits the classification into the exceptional values δ=0,1,2 and the generic case. For each case the paper gives a complete list of pairwise non-isomorphic algebras, using the automorphism group of $μ_n^{0}$ to normalize parameters. It also constructs all δ-Poisson structures on $μ_n^{0}$ and shows they are all trivial, meaning the Lie bracket is forced to vanish. If the classification is correct, the transposed δ-Poisson world on this family is fully understood and the only special behavior occurs at the roots of δ(δ−1)(δ−2).

What carries the argument

The central object is the n-dimensional null-filiform associative algebra $μ_n^{0}$: a basis e_1,...,e_n with products e_i·e_j=e_{i+j} and all other products zero. Because the algebra is generated by e_1, identity (2) read on triples of basis elements yields a recurrence for [e_1,e_{i+1}] in terms of e_1·[e_1,e_i] and e_{i−1}·[e_1,e_2], and the consistency of this recurrence produces the polynomial δ(δ−1)(δ−2) as the switch separating the exceptional cases. The other carrying mechanism is the automorphism group of $μ_n^{0}$, given by triangular coordinate changes with A_1≠0, which acts on the parameters α_t in [e_1,e_2]; the isomorphism relations derived in Theorems 12, 20, and 27 turn parameter lists into the displayed non-isomorphic normal forms.

What would settle it

Using the formulas of Corollary 10, enumerate all Lie brackets on $μ_5^{0}$ satisfying identity (2) for δ=0,1,2,3 with a computer algebra system and compare them with the normal forms in Theorems 18, 25, 8, and 28; any bracket not isomorphic to a listed representative would disprove the completeness claim, and for δ=2 this directly tests the quoted classification [5].

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Extended reading notes

Core claim

The paper establishes that the Lie brackets making ($μ_n^{0}$, ·) a transposed δ-Poisson algebra are fully controlled by δ. The recurrence derived from identity (2) gives [e_1,e_3]=δ e_1·[e_1,e_2], [e_1,e_4]=($δ^{2}$+δ)/2 e_2·[e_1,e_2], and [e_1,e_5]=($δ^{3}$+$δ^{2}$+2δ)/4 e_3·[e_1,e_2]; comparing this last expression with the triple {e_2,e_1,e_3} forces ($δ^{3}$−$3δ^{2}$+2δ)e_3·[e_1,e_2]=0. Hence, for δ outside {0,1,2}, the bracket is concentrated in the top degrees, with [e_1,e_2]=α_{n−2}e_{n−2}+α_{n−1}e_{n−1}+α_n e_n and [e_1,e_i]=0 for i≥5, while δ=0 and δ=1 each have their own infinite families and δ=2 is the classical transposed Poisson case taken from [5]. The paper classifies each family up to isomorphism, handling dimensions 2, 3, and 4 separately, and proves in Theorem 34 that every δ-Poisson structure on $μ_n^{0}$ is trivial.

Load-bearing premise

The load-bearing premise is that the previously reported classification for δ=2, quoted from [5] as Theorem 8, is complete and correct, because this paper's own computations cover only δ=0, δ=1, and δ outside {0,1,2}.

Editorial extensions

If this is right

  • For δ outside {0,1,2}, every transposed δ-Poisson bracket is supported only in degrees n−2, n−1, n, with [e_1,e_i]=0 for i≥5, so the structures form a three-parameter family with at most six isomorphism classes.
  • For δ=0, every structure has only [e_1,e_2] nonzero, and up to isomorphism only the normal forms of Theorem 18 occur.
  • For δ=1, the bracket is encoded by [e_1,e_i]=Σ_{t=i}^n α_{t−i+2}e_t for i≥2, with α_1=0 forced by the Jacobi identity, giving the normal forms of Theorem 25.
  • Theorem 34 implies that no nontrivial δ-Poisson structure exists on μ_n^0: the compatibility identity plus the one-generated nature of the algebra forces every bracket to vanish, in contrast with the transposed case.
  • The δ=2 case recovers the classical transposed Poisson classification quoted from [5] and shows it fits the same δ-pattern established for the other values.

Reading between the lines

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

  • Beyond the paper: the same recurrence-and-obstruction method should apply to other one-generated nilpotent associative algebras, where a higher-degree polynomial in δ is likely to play the role of δ(δ−1)(δ−2).
  • Beyond the paper: because the generic regime has only finitely many isomorphism classes, the δ outside {0,1,2} families give a natural supply of rigid examples for testing deformation and degeneration questions in transposed δ-Poisson algebras.
  • Beyond the paper: the triviality result for δ-Poisson structures suggests that the two compatibility identities (1) and (2) have very different rigidity on nilpotent associative algebras; a testable extension is whether other singly generated nilpotent associative algebras also admit only trivial δ-Poisson brackets.
  • Beyond the paper: the δ=2 branch could be independently checked by applying the paper's own isomorphism relations to the formulas of Corollary 10; a mismatch there would require revision of the completeness claim.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies transposed δ-Poisson algebra structures on the n-dimensional null-filiform associative algebra μ_n^0 over C. Section 3 derives the polynomial condition δ^3−3δ^2+2δ=0 from the defining identity, and then presents classifications: Theorem 18 for δ=0, Theorem 25 for δ=1, and Theorem 28 for δ∉{0,1,2}, together with low-dimensional cases in Theorems 29–33. The δ=2 case is imported from the overlapping-author preprint [5] via Theorem 8. Theorem 34 asserts that all δ-Poisson algebra structures on μ_n^0 are trivial. The paper's headline claim is a complete classification of transposed δ-Poisson structures for every value of δ, with the non-generic values characterized by the roots of the displayed cubic.

Significance. If the gaps identified below are repaired, this is a useful contribution to the classification program for transposed δ-Poisson algebras: the δ=0, δ=1, and generic-δ results are derived directly from the defining identity, the normal forms are explicit, and the isomorphism equations in Theorems 12, 20, and 27 are concrete and checkable. The theorem that all δ-Poisson structures on μ_n^0 are trivial is simple and clean. However, the completeness claim for all δ currently rests on an unverified import from [5], and the generic classification in Theorem 28 has a concrete gap for dimension n=5. These issues affect the central advertised result, not just presentation.

major comments (3)
  1. [Section 3, Theorem 28, proof case (1)] The normalization of α_{n-2} to 1 by choosing A_1 = α_{n-2}^{1/(n-5)} is impossible for n=5. According to the transformation formula in Theorem 27, α'_{n-2}=α_{n-2}/A_1^{n-5}; for n=5 this gives α'_3=α_3 for every automorphism, so α_3 is an invariant of the algebra TP_δ(α_3,α_4,α_5). Consequently the listed normal forms TP_δ(1,0,0), TP_{(n-2)/2}(1,α,0), TP_{δ^2+3δ=2n-4}(1,0,α) do not cover the orbits with α_3≠1 in dimension 5. Theorem 28 is therefore false as stated for n=5, and the dimension-5 case requires a separate treatment, e.g. a continuous nonzero parameter α_3 in the relevant families.
  2. [Section 2, Theorem 8 and the abstract] The abstract claims a complete classification for every δ, but the δ=2 case is not proved or independently checked in this paper. The classification of transposed Poisson structures, i.e. δ=2, is quoted as Theorem 8 from the overlapping-author preprint [5], with Theorem 6 giving the general form; no derivation or verification is included. The paper's own Section 3 covers δ=0 (Theorem 18), δ=1 (Theorem 25), and δ∉{0,1,2} (Theorem 28), and Theorem 26 explicitly excludes δ=2. Since an error or omission in [5] would invalidate the headline completeness claim, the paper should either prove Theorem 8 or clearly state that the δ=2 part is imported and remains conditional on [5].
  3. [Section 3, Theorems 18, 25, 28] The pairwise non-isomorphism assertions in the classification theorems are not established. The proofs show that every algebra is isomorphic to one of the listed normal forms, but they do not prove that the listed families are pairwise non-isomorphic or that the continuous parameters are invariants. The missing checks are available from the isomorphism formulas in Theorem 27 (for example α_{n-1} is invariant when δ=(n-2)/2, and α_n is invariant when δ=(n-1)/2), but they should be stated and verified explicitly. The parameter domains also need to be fixed: in Theorem 25 the family TP_1(1,0,...,0,α_s,0,...,0) with α_s=0 coincides with TP_1(1,0,...,0) for every s, so either α_s should be restricted to C^* and the algebra TP_1(1,0,...,0) listed separately, or the duplication must be addressed; the same care is needed for the exceptional families in Theorem 28.
minor comments (4)
  1. [Abstract] The first sentence says transposed δ-Poisson algebras are 'a generalization of transposed δ-Poisson algebras', which is circular and presumably should read 'a generalization of transposed Poisson algebras'; the phrase 'It can be seen that these algebras are characterized by the roots...' is also vague and should be replaced by a precise statement.
  2. [Section 3, Theorem 28] The notation 'TP n−2 2 (1,α,0)' and 'TP δ2+3δ=2n−4(1,0,α)' is unconventional and hard to parse; the parameter δ should be specified by explicit equations such as δ=(n−2)/2 or δ^2+3δ=2n−4.
  3. [Section 3, Theorem 26 proof] The sentence 'Applying induction by i+j, we prove [e_i,e_j]=0 for i+j≥6' is too terse: the displayed argument with the triple {e_1,e_i,e_j} only gives [e_{i+1},e_j]=0 in certain ranges, and the remaining triangular cases should be spelled out.
  4. [Throughout] There are several small typographical issues, such as 'multimplication rules' in Lemmas 22–24, and inconsistent use of C versus C^* in parameter ranges; these should be cleaned up in a revision.

Circularity Check

1 steps flagged · score 4.0 of 10

The abstract's complete classification for every δ depends on the δ=2 case imported from the overlapping-author preprint [5]; the paper's own theorems cover δ=0, δ=1, and δ outside {0,1,2}.

  1. self citation load bearing [Abstract; Section 2, Theorems 6 and 8; Section 3, Theorem 26]
    "In [5], all transposed Poisson algebra structures on null-filiform associative algebras were completely classified. ... Theorem 8. Let (µ^0_n,·,[−,−]) be a transposed Poisson algebra and n≥5. Then this algebra is isomorphic to one of the following pairwise non-isomorphic algebras: TP2(1,0,...,0), TP2(0,α,0,...,0), TP2(0,...,0,1_s,0,...,0,α_{2s−3},0,...,0). ... Theorem 26. Let (µ^0_n,·,[−,−]) be a transposed δ-Poisson algebra structure defined on the associative algebra µ^0_n, n≥5 and δ≠0,1,2."

    The paper's own classification theorems cover δ=0 (Theorem 18), δ=1 (Theorem 25), and δ∉{0,1,2} (Theorems 26 and 28). The remaining case δ=2, n≥5, is the transposed Poisson case, and it is dispatched by quoting Theorem 8 from the overlapping-author preprint [5]; Theorem 26 explicitly excludes δ=2, and Corollary 10 gives only necessary restrictions for δ=2, not a classification. The abstract claims 'a complete classification ... corresponding to each value of the parameter δ', but for δ=2 that completeness is imported from a same-author citation that is neither reproved nor independently checked in the present paper, so the headline classification is partly forced through the self-citation rather than by the paper's own derivations.

full rationale

The new derivations for δ=0, δ=1, and the generic case δ∉{0,1,2} are self-contained: starting from the defining identity (2), the null-filiform multiplication e_i·e_j=e_{i+j}, and the automorphism description in Theorem 5, the paper derives bracket restrictions (Theorem 9/Corollary 10), obtains the families TP0, TP1, TPδ (Theorems 11, 19, 26), and computes isomorphism relations (Theorems 12, 20, 27). No parameter is fitted and no empirical prediction is being reimported. The genuine circularity concern is concentrated in the δ=2 case for n≥5: the paper's own theorems exclude that case, and the classification is taken from the overlapping-author preprint [5] as Theorem 8 without proof or independent verification. This makes the abstract's completeness claim partly load-bearing on a self-citation, though the substantial independent content for the other δ-values prevents the whole paper from being circular. The unproved pairwise non-isomorphism assertion in Theorem 28 is a proof gap rather than a circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are introduced in the derivation; the α_i coefficients are the structure constants being classified, not fitted inputs. The paper's computations rely on standard background: the model of null-filiform algebras, the automorphism group from [6], and the δ=2 classification from [5]. No new entities are postulated.

assumptions (4)
  • domain assumption The only n-dimensional null-filiform associative algebra over C is μ_n^0 with e_i·e_j = e_{i+j}
    Quoted from [29, Proposition 5.3] as Theorem 4.
  • domain assumption Automorphisms of μ_n^0 take the triangular polynomial form of Theorem 5
    Quoted from [6] as Theorem 5; used in all normalization lemmas.
  • domain assumption The classification of transposed Poisson (δ=2) algebras on μ_n^0 given in [5] is complete
    The paper cites Theorems 6-8 from [5] to cover the δ=2 case in the abstract's claim of completeness.
  • domain assumption The base field is C with characteristic 0
    Stated at the start of Section 2; used for division by integers like 2 and for the polynomial identity to imply each factor vanishes.

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Pith. "Pith review of Transposed $\delta$-Poisson algebra structures on null-filiform associative algebras." pith.science (2026). https://pith.science/paper/2RHMYC5G

@misc{pith2026250710554,
  author       = {Pith},
  title        = {Pith review of: Transposed $\delta$-Poisson algebra structures on null-filiform associative algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2RHMYC5G}},
  note         = {Machine review of arXiv:2507.10554}
}
abstract

In this paper, we consider transposed $\delta$-Poisson algebras, which are a generalization of transposed $\delta$-Poisson algebras. In particular, we describe all transposed $\delta$-Poisson algebras of associative null-filiform algebras. It can be seen that these algebras are characterized by the roots of the polynomial $\delta^3 - 3\delta^2 + 2\delta$. A complete classification of transposed $\delta$-Poisson algebras corresponding to each value of the parameter $\delta$ is provided. Furthermore, we construct all $\delta$-Poisson algebra structures on null-filiform associative algebras, and show that they are trivial $\delta$-Poisson algebras.

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