Biderivations, local and 2-local derivation and automorphism of simple $\omega$-Lie algebras

Hassan Oubba

REVIEW 5 major objections 5 minor 29 references

Biderivations, local and 2-local derivation and automorphism of simple $\omega$-Lie algebras

T0 review · 5 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read For finite-dimensional simple ω-Lie algebras, local and 2-local derivations are global and the same rigidity holds for automorphisms, biderivations, and 1/2-derivations.

desk verdict The paper's central biderivation theorem is false for the non-Lie simple ω-Lie algebras, contradicting the paper's own cited classification; the 4-dimensional table and 1/2-derivation computations are worth preserving but the paper as written needs major revision. read the letter →

arxiv 2505.00436 v1 pith:WNVXWTDG submitted 2025-05-01 math.RA

Hassan Oubba This is my paper · ORCID

classification math.RA MSC 17A3217B3215A9917B6017A30

keywords ω-Liealgebraslocalderivations2-localautomorphismsbiderivations1/2-derivationssemisimple

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 sets out to show that rigidity results familiar for finite-dimensional simple Lie algebras survive when the Jacobi identity is relaxed by a skew 2-form, in the class known as ω-Lie algebras. Concretely, it claims that on every finite-dimensional semisimple complex ω-Lie algebra, a linear map that agrees with a derivation at each point is itself a derivation, and the same holds for 2-local derivations; automorphisms obey a similar pointwise-to-global principle, with local automorphisms allowed to be anti-automorphisms. It also claims a complete description of biderivations, which are exactly scalar multiples of the bracket on simple algebras, and of 1/2-derivations, which are exactly scalar maps. The argument is carried by low-dimensional classifications of ω-Lie algebras together with explicit matrix tables of derivations and automorphisms, so the paper is as much a computational toolkit as a rigidity theorem.

What carries the argument

The working object is the ω-Lie algebra: a vector space with a skew-symmetric bilinear bracket and a skew-symmetric bilinear form ω satisfying the ω-Jacobi identity $$[[x,y],z]+[[y,z],x]+[[z,x],y]=\omega(x,y)z+\omega(y,z)x+\omega(z,x)y.$$ The proof machinery is low-dimensional classification plus explicit matrix control: the 3- and 4-dimensional ω-Lie algebras are known up to isomorphism, and their derivations and automorphisms are written out as parameterized matrices. A local or 2-local map is matched against these matrices point by point, and the matching forces the map's own matrix to have exactly the global form. Biderivations are treated by splitting into symmetric and skew-symmetric parts, converting the symmetric part into an anti-commuting linear map and the skew-symmetric part into a commuting map in the centroid; the non-Lie simple cases A_α, B, and C_α are handled by direct coefficient comparison. For 1/2-derivations the same direct comparison on those simple algebras shows that only scalar matrices survive.

What would settle it

Evaluate the first biderivation equation for δ(x,y)=[x,y] in A_α at (e_1,e_2,e_3). The left side is [e_1,e_3]=e_1+e_2, while the right side is [e_1,[e_2,e_3]]+[[e_1,e_3],e_2]=(e_1+e_2)+e_1=2e_1+e_2; these differ by -e_1, which is exactly ω(e_2,e_3)e_1 with ω(e_2,e_3)=-1. Thus the bracket itself is not a biderivation of A_α, so the claimed characterization cannot hold for that algebra as stated.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that pointwise data determine global algebraic structure for finite-dimensional complex ω-Lie algebras. A local derivation, meaning a map that agrees with some derivation on each element, must itself be a derivation; a 2-local derivation, which agrees with a possibly different derivation on each pair, must also be a derivation. The analogous statement for automorphisms is that a local automorphism is an automorphism or an anti-automorphism, and a 2-local automorphism is an automorphism. For biderivations the claimed description is δ(x,y)=λ[x,y] for a fixed scalar λ, and for 1/2-derivations it is Δ(x)=λx. In the 4-dimensional case the paper gives explicit parameter lists for all symmetric biderivations and proves that skew-symmetric biderivations vanish.

Load-bearing premise

The biderivation theorem for non-Lie simple ω-Lie algebras assumes that the map (x,y)↦λ[x,y] satisfies the biderivation equations, which would require the bracket to obey the ordinary Jacobi identity instead of the ω-corrected identity that actually defines these algebras.

Editorial extensions

If this is right

For any finite-dimensional semisimple complex ω-Lie algebra, the local and 2-local derivation spaces coincide with the ordinary derivation algebra, so no new pointwise-defined derivations appear outside Der(g).
On finite-dimensional simple ω-Lie algebras, biderivations form a one-dimensional space spanned by the bracket; in particular, the only symmetric biderivations of a simple Lie algebra are zero.
Every 2-local automorphism of such an algebra is an automorphism, while every local automorphism is an automorphism or an anti-automorphism, so local data cannot distinguish the automorphism group from its anti-involutive counterpart.
The explicit table of symmetric biderivations for the 4-dimensional complex ω-Lie algebras, together with the vanishing of skew-symmetric biderivations, gives a full description of BDer(g) in dimension four.
Every 1/2-derivation of a finite-dimensional simple complex ω-Lie algebra is multiplication by a scalar, and every local or 2-local 1/2-derivation is the same global scalar map.

Reading between the lines

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

Editorial extension: the coefficient-comparison method used in Section 7 could be run on every algebra in the 4-dimensional classification, yielding a complete table of 1/2-derivations that the paper leaves implicit.
Editorial extension: because Theorem 3.3 reduces semisimple ω-Lie algebras to the Lie case and dimension at most four, an automated implementation of the matrix-matching argument is a natural next step and would make the local/2-local rigidity check routine for dimension five and above.
Editorial extension: the closing question on transposed Poisson structures for ω-Lie algebras points to a testable consequence—if 1/2-derivations are only scalar maps, any reasonable ω-analogue of a transposed Poisson structure would also be scalar, so new structure would have to come from the 2-form itself rather than from linear maps.

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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

You should know two things about arXiv:2505.00436. First, there is a genuine computational contribution in the 4-dimensional symmetric biderivation table and the 1/2-derivation classification for the 3-dimensional non-Lie simple ω-Lie algebras. Second, the paper's flagship Theorem 6.3 is false as stated, and the error is not a typo.

The paper claims to characterize biderivations of finite-dimensional simple ω-Lie algebras over C by δ(x,y)=λ[x,y]. For the non-Lie simple algebras Aα, B, Cα, that fails. Take the algebra B from Theorem 3.1: [e1,e2]=e2, [e1,e3]=e2+e3, [e2,e3]=e1. If δ=λ[·,·], the first biderivation identity requires λ[[e1,e2],e3] = λ([e1,[e2,e3]]+[[e1,e3],e2]), i.e. λe1 = -λe1. Only λ=0 works. The paper itself says just before Theorem 6.3 that the author's previous paper [23] proved all biderivations of Aα, B, Cα are zero. So the theorem contradicts its own cited classification. The correct statement would separate the Lie case (λ[x,y]) from the non-Lie simple case (only zero). This is load-bearing: the abstract advertises a full biderivation characterization.

What is good: Table 1 for symmetric biderivations of the 4-dimensional ω-Lie algebras looks like honest, checkable computation, and I don't know another source for it. The 1/2-derivation theorem for Aα, B, Cα is also new as far as I can tell, and the worked equations are consistent. The local/2-local derivation and automorphism results for the 4-dimensional algebras are built on the known tables from Chen–Zhang et al., and the pattern is plausible, though most cases are dismissed with 'similar arguments' and only one or two exemplars are worked out in detail.

Secondary problems: the paper leans on the author's own [23] for the 3-dimensional cases, which is fine as self-citation, but the contradiction above makes the reliance fatal. There are broken references (the unnumbered [?] in Lemma 2.2, 'Theorem ??' in Section 7), a missing symbol in the abstract, and several typos. These are minor by comparison.

Who is this for? Someone working on local/2-local maps or biderivations in non-associative algebras might find the corrected computational results useful. But the paper cannot be trusted until Theorem 6.3 is fixed and the omitted proofs are supplied. My recommendation: if you are refereeing it, reject in current form but communicate that the computational tables and the 1/2-derivation analysis are worth preserving. If you are an editor, send it to a referee who can verify the table, but expect major revision.

Referee Report

5 major / 5 minor

Summary. The paper studies finite-dimensional complex simple ω-Lie algebras. It claims that every local and 2-local derivation is a derivation, every local automorphism is an automorphism or anti-automorphism, every 2-local automorphism is an automorphism, that biderivations are exactly the multiples λ[x,y] of the bracket, and that all ½-derivations are scalar multiples of the identity. The arguments are computational and rely on classifications of 3- and 4-dimensional ω-Lie algebras and on the author's previous paper [23]. The main asserted results are presented for all finite-dimensional simple ω-Lie algebras, with special attention to the low-dimensional non-Lie examples.

Significance. If correct, these results would extend the classical theory of local and 2-local maps and of biderivations from simple Lie algebras to ω-Lie algebras, which appear in geometric contexts. The claimed characterization of biderivations and of ½-derivations would be a useful contribution. However, the central biderivation theorem is false for non-Lie simple ω-Lie algebras, and several proofs are incomplete or assume linearity. The paper also relies substantially on the author's own earlier work [23] for the 3-dimensional cases, without providing independent verification. These issues undermine the validity of the main claims.

major comments (5)

[Theorem 6.3] Theorem 6.3 is false as stated: the bilinear map δ(x,y)=λ[x,y] is not a biderivation on non-Lie simple ω-Lie algebras for λ≠0. Substituting δ=λ[·,·] into the first identity of Definition 6.1 reduces the condition to the Jacobi identity [[x,y],z]+[[y,z],x]+[[z,x],y]=0, while Definition 3.1 replaces this by the ω-corrected identity whose right-hand side is generally nonzero. For example, take the algebra B of Theorem 3.1(4) with [e2,e3]=e1 and ω(e2,e3)=2. Then for x=e2, y=e3, z=e1, the left side of the first biderivation identity is λ[[e2,e3],e1]=0, while the right side is λ([e2,[e3,e1]]+[[e2,e1],e3])=-2λe1. Hence the claimed equivalence fails already in dimension 3. The paper itself states immediately before Theorem 6.3 that the biderivations of Aα, B, and Cα are zero, so the theorem also contradicts its own cited result [23].
[Theorem 4.1, proof] The proof of Theorem 4.1 assumes without justification that the local derivation Δ is linear: it writes Δ(x)=B x̄ for a single matrix B and then compares Δ(x) with D_x(x). A local derivation is defined pointwise, with a possibly different derivation D_x for each x, and linearity is not given. The coefficient equations derived from this comparison therefore presuppose what needs to be proved. The same issue appears in the proofs of Theorems 5.1 and 5.3 for local and 2-local automorphisms, where a global matrix B is introduced before linearity or even well-definedness of the map as a single linear transformation is established.
[Theorems 4.2, 4.4, 5.1, 5.3, 6.1, 6.2] Several theorems are stated for broad classes of algebras but are proven only for one or a few algebras, with the remaining cases dismissed by 'similar arguments' or 'analogously'. For example, Theorem 4.2 claims every local derivation of any 4-dimensional ω-Lie algebra is a derivation, but only L1,1 is treated; Theorem 6.1 claims no nontrivial skew-symmetric biderivations for all 4-dimensional ω-Lie algebras but only L1,6 is shown; and Theorem 6.2 claims a full description of symmetric biderivations on L1,1 but does not verify that the displayed formula actually satisfies the biderivation identities. Since the central theorems for simple algebras (Theorems 4.3, 4.5, 5.2, 5.4) rely on these case-by-case results, the omitted cases are load-bearing and the proofs are incomplete.
[Theorem 7.1, equations (7.1)–(7.9)] The proof of Theorem 7.1 contains concrete computational errors. For the algebra Aα, equation (7.1) states 2a11 = a11 −αa31 + a22, but direct computation of [Δ(e1),e2]+[e1,Δ(e2)] yields an additional term a32 in the coefficient of e1; the omitted term affects the conclusion. Similarly, for the algebra B, equation (7.4) states 2a11 = −a31, whereas the coefficient of e1 in 2Δ(e2) is 2a12, so the displayed equation should be 2a12 = −a31. Thus the system of equations used to derive Δ=λI is not correctly written, and the conclusion is not established by the given argument.
[Section 6, decomposition of BDer(g)] The paper asserts without proof that for any biderivation δ the symmetric part δ+ and skew-symmetric part δ− are again biderivations, so that BDer(g)=BDer+(g)⊕BDer−(g). For Lie algebras this follows from the Jacobi identity, but for a general ω-Lie algebra the map δ^T(x,y)=δ(y,x) need not satisfy the two identities of Definition 6.1. This decomposition is used to reduce the biderivation problem to symmetric and skew-symmetric cases, so the missing justification is a gap in the method; the later counterexample to Theorem 6.3 shows the danger of transferring Lie-algebra facts to ω-Lie algebras.

minor comments (5)

[Throughout] There are numerous typos and grammatical errors, e.g., 'automorphis', 'qutomorphism', 'bideivation', 'Tble', 'that it' for 'that is', and 'similary' variants. These should be corrected.
[Sections 4–5] References to 'Theorem ??' in Section 7 and to 'Table.4 in [10]' without a table number in the reference list make it hard for the reader to verify the cited derivation and automorphism classifications.
[Section 4, equation (4.3)] The line 'b11 = b12 = b21 = b22 = b31 = b32 = b41 = b42' appears to be a typographical error; the coefficient comparison actually forces these entries to be zero, not merely equal to one another.
[Theorem 5.3, proof] In the proof for L2,3, the maps ψ_{x,e4} are called 'derivations' but should be 'automorphisms'; this is a terminology slip that does not affect the argument.
[Section 1, Introduction] The introduction contains an incomplete sentence: 'In [23], we give a description of local (2-local) derivations and automorphisms and biderivations of three dimensional complex ω-Lie algebras in this paper we give similar result for four dimensional complexω-Lie algebras' — this needs to be split and clarified.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction by construction; the only mild concern is reliance on the author's own prior 3D classification, which is a self-citation dependency rather than an input-output equivalence.

full rationale

The paper's original 4D computations are direct matrix arguments: Theorems 4.1, 4.4, 5.1, 5.3 and the biderivation calculations in Section 6 substitute the derivation/automorphism matrices from the external tables in [10] and read off the required form of the local map, so the conclusions are not assumed in the setup. The Lie-algebra cases are cited to external theorems ([2], [3], [9], [12], [26]). The 3-dimensional non-Lie simple cases are cited to the author's own earlier paper [23]. This is a genuine self-citation and it is load-bearing for the 3D part of the simple-algebra statements, but it is not a circularity in the sense of this review: it is a reference to a separate published classification, not a fitted parameter renamed as a prediction, and no equation is made true by definition. There are also omitted proofs for many 4D cases and missing reference markers such as 'Theorem ??' in Theorem 7.2, but those are completeness defects, not circularity. Finally, Theorem 6.3 is mathematically overbroad: for non-Lie simple omega-Lie algebras such as B the map lambda[.,.] with lambda != 0 is not a biderivation because the ordinary Jacobi identity fails, so the 'if' direction is false. That is a correctness defect in the inference from [23]'s zero-biderivation statement, not a circular input-output identity.

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

No new physical or algebraic entities are introduced. The proofs rest entirely on prior classifications and tables, including the author's own [23]. The constants in the biderivation table are legitimate degrees of freedom in the answer, not parameters tuned to force a result.

assumptions (5)

domain assumption The classifications of 3- and 4-dimensional complex omega-Lie algebras given in [8] and [7] are correct.
Used to enumerate all cases in Sections 4-7.
domain assumption The derivation and automorphism tables for 4-dimensional omega-Lie algebras from [10] are complete and correct.
Used as black boxes in local/2-local and biderivation proofs.
domain assumption Known results that local and 2-local derivations/automorphisms of finite-dimensional simple Lie algebras are global ([2], [3], [9], [12]).
Used to handle the Lie-algebra case of the main theorems.
domain assumption The author's earlier paper [23] correctly proves the 3-dimensional omega-Lie algebra analogues.
Theorems 4.3, 4.5, 5.2, 5.4 and 6.3 depend on these claims.
domain assumption Proposition 1.5 of [7]: every derivation of a 4-dimensional omega-Lie algebra except L1,6 and L1,8 is an omega-derivation.
Used in Theorem 6.4.

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Cite this review

Pith. "Pith review of Biderivations, local and 2-local derivation and automorphism of simple $\omega$-Lie algebras." pith.science (2026). https://pith.science/paper/WNVXWTDG

@misc{pith2026250500436,
  author       = {Pith},
  title        = {Pith review of: Biderivations, local and 2-local derivation and automorphism of simple $\omega$-Lie algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WNVXWTDG}},
  note         = {Machine review of arXiv:2505.00436}
}

abstract

Given a finite-dimensional complex simple $\omega$-Lie algebras $\mathfrak{}$ over $\mathbb{C}$. We prove that every local ,$2-$local derivation is a derivation and every local (resp. 2-local) automorphisms are automorphisms or an anti-automorphis (resp. automorphism). We characterize also biderivation, $\frac{1}{2}$-derivation and local (2-local) $\frac{1}{2}$-derivation of $\mathfrak{g}$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [23]

Oubba, H. (2024). Local (2-Local) derivations and auto morphisms and biderivations of complex ω-Lie algebras. Le Matematiche, 79(1), 135-150

work page 2024

[2] [1]

Kudaybergenov, K

Ayupov, Sh. Kudaybergenov, K. 2-Local automorphisms on ﬁnite-dimensional Lie algebras. Linear Algebra and its Applications 507 (2016) 121 −131

work page 2016

[3] [2]

Kudaybergenov, K

Ayupov, Sh. Kudaybergenov, K. Local derivations on ﬁnit e-dimensional Lie algebras. Linear Algebra and its Ap- plications 493 (2016) 381 −398

work page 2016

[4] [3]

”2-Local derivations on ﬁnite- dimensional Lie algebras.” Linear Algebra and its Applicat ions 474 (2015): 1-11

Ayupov, Shavkat, Karimbergen Kudaybergenov, and Isami ddin Rakhimov. ”2-Local derivations on ﬁnite- dimensional Lie algebras.” Linear Algebra and its Applicat ions 474 (2015): 1-11

work page 2015

[5] [4]

Breˇ sar, M., Martindale 3rd, W.S., Miers, C.R.: Central izing maps in prime rings with involution. J. Algebra. 161, (1993) 342−357

work page 1993

[6] [5]

Breˇ sar, M., Zhao, K.: Biderivations and commuting Line ar maps on Lie algebras. J. Lie Theory. 28, (1018) 885−900. 17

work page

[7] [6]

Taiwanese J

Breˇ sar, M.: Commuting maps: A survey. Taiwanese J. Math . 8, (2004)361−397

work page 2004

[8] [7]

To appear in Bull

Chen Y ., Zhang R., Simple ω-Lie algebras and 4-dimensional ω-Lie algebras over C. To appear in Bull. Malays

work page