An Algorithmic Approach to Inner Derivations of Low-Dimensional Zinbiel Algebras

Ahmed Zahari; Bouzid Mosbahi

REVIEW 4 major objections 4 minor 1 cited by

An Algorithmic Approach to Inner Derivations of Low-Dimensional Zinbiel Algebras

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that every element w of a Zinbiel algebra defines an inner derivation by the commutator map u ↦ w∘u − u∘w, with explicit matrices in low dimensions.

desk verdict The paper's central claim is false: the maps ad_w are not derivations, as the paper's own A1_4 example shows. read the letter →

arxiv 2412.20599 v1 pith:XIHEGE45 submitted 2024-12-29 math.RA

Bouzid Mosbahi , Ahmed Zahari This is my paper · ORCID

classification math.RA MSC 16D7017A3017A32

keywords Zinbielalgebrasinnerderivationscommutatormapslow-dimensionalcomplexmatrixalgorithmstructureconstantsLeibniz

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

Zinbiel algebras are vector spaces with a bilinear product subject to $(u\circ v)\circ w=u\circ(v\circ w)+u\circ(w\circ v)$, and they sit in duality with Leibniz algebras. This paper tries to establish that every element $w$ of such an algebra gives an inner derivation, defined as the commutator map $\mathrm{ad}_w(u)=w\circ u-u\circ w$, and that these maps satisfy the Leibniz rule. It then turns that definition into an algorithm that writes $\mathrm{ad}_w$ as a matrix from the algebra's structure constants, and it applies the algorithm to all complex Zinbiel algebras of dimensions two, three, and four, producing explicit tables. If the characterization stands, the tables give a computable, isomorphism-invariant dimension for the inner-derivation space of each listed algebra.

What carries the argument

The engine of the construction is the commutator bracket $[u,v]=u\circ v-v\circ u$ and the associated map $\mathrm{ad}_w(u)=w\circ u-u\circ w$. The derivation check is the Leibniz-rule computation for $\mathrm{ad}_w$ on a product $u\circ v$: the left side $\mathrm{ad}_w(u\circ v)$ and the right side $\mathrm{ad}_w(u)\circ v+u\circ \mathrm{ad}_w(v)$ are expanded using bilinearity and compared. The proof invokes the Zinbiel identity in the form $(u\circ v)\circ w=u\circ(v\circ w)+u\circ(w\circ v)$; the algorithmic part then reduces the computation to structure constants, giving $d_{ij}=\sum_t a_t\gamma^j_{it}-\sum_t a_t\gamma^j_{ti}$.

What would settle it

For the algebra $A^1_4$ in Theorem 4.5, the multiplication table gives $e_1\circ e_1=e_2$, $e_1\circ e_2=e_3$, $e_2\circ e_1=2e_3$, $e_1\circ e_3=e_4$, $e_2\circ e_2=3e_4$, and $e_3\circ e_1=3e_4$. Taking $w=e_1$, $u=e_1$, $v=e_2$, the Leibniz left side $\mathrm{ad}_w(e_1\circ e_2)=\mathrm{ad}_w(e_3)=-2e_4$, while the right side $\mathrm{ad}_w(e_1)\circ e_2+e_1\circ \mathrm{ad}_w(e_2)=0+e_1\circ(-e_3)=-e_4$. The two sides differ, so the claimed derivation property fails on this listed algebra.

Watch

Extended reading notes

Core claim

The central claim is Proposition 3.5: for any Zinbiel algebra $A$ and any $w\in A$, the map $\mathrm{ad}_w(u)=w\circ u-u\circ w$ is a derivation of $A$ in the sense of Definition 2.3. On the paper's terms, this makes the set $\mathrm{Inn}(A)$ of all such maps an ideal of the Lie algebra $\mathrm{Der}(A)$, and it justifies treating these maps as the 'inner' derivations of the algebra. The accompanying algorithm computes the matrix of $\mathrm{ad}_w$ with respect to a basis: if $w=\sum_t a_t e_t$ and the product is encoded by structure constants $\gamma^j_{it}$, then the $(i,j)$ entry is $d_{ij}=\sum_t a_t\gamma^j_{it}-\sum_t a_t\gamma^j_{ti}$. Applying this recipe to the classifications listed in the paper yields the matrix tables in Theorems 4.2, 4.4, and 4.6, together with dimension ranges for the inner-derivation spaces.

Load-bearing premise

The argument that $\mathrm{ad}_w$ is a derivation depends on treating the Zinbiel product as associative for the purpose of the proof; the defining identity actually contains an additional term $u\circ(w\circ v)$ that the proof does not carry along.

Editorial extensions

If this is right

Under the paper's claims, the dimension of the inner-derivation space is an isomorphism invariant, and the tables record values ranging from zero to three across the classified algebras.
The matrix algorithm is uniform in the dimension, so any $n$-dimensional Zinbiel algebra with known structure constants can be fed into the same procedure to obtain its inner-derivation matrices.
Because Theorem 2.5 carries every derivation to the associated Lie algebra, the inner-derivation tables also constrain possible Lie-algebra derivations on these Zinbiel algebras.
The listed tables separate the isomorphism classes by inspection: classes with different inner-derivation dimensions are non-isomorphic.

Reading between the lines

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

The matrix formula $d_{ij}=\sum_t a_t\gamma^j_{it}-\sum_t a_t\gamma^j_{ti}$ computes the commutator map for any bilinear product, so the algorithm remains usable even if the derivation property is examined separately; the tables can be read as descriptions of commutator maps in their own right.
For an associative algebra the same commutator map is a genuine derivation, so this algorithmic treatment would specialize to the classical inner-derivation matrices of associative algebra theory.
Running the same derivation check on each listed four-dimensional class would show exactly where the defining identity's extra term matters and which classes satisfy the proposed derivation property.
The same templated computation could be repeated for Leibniz, dendriform, or other non-associative products by substituting the corresponding defining identity.

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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

Plainly: the central claim is false. The stress-test counterexample is valid and uses the paper's own algebra A1_4. With the product from Theorem 4.5 (e1∘e2=e3, e2∘e1=2e3, e1∘e3=e4, e3∘e1=3e4), take w=e1. Then ad_{e1}(e3)=e3∘e1−e1∘e3=3e4−e4=2e4, but the Leibniz rule for a derivation d gives d(e1∘e2)=d(e1)∘e2 + e1∘d(e2)=0+e1∘e3=e4. 2e4≠e4, so ad_{e1} is not a derivation. The proof of Proposition 3.5 simply drops the second term of the Zinbiel identity: Definition 2.1 is (u∘v)∘w=u∘(v∘w)+u∘(w∘v), but the proof uses (u∘v)∘w=u∘(v∘w). That is not a minor slip; it is exactly the term that prevents the Leibniz rule from holding.

What is new here is only the definition of ad_w for Zinbiel algebras, and that definition does not produce derivations. The algorithm in Section 4 is a restatement of how to compute the matrix of a linear map in a basis, correct but trivial. The tables are routine computations from known classifications, and they inherit the central error. There are also independent sloppiness issues: Lemma 2.6 claims left/right multiplication operators are derivations, which is generally false; Lemma 3.4 uses the same wrong identity; Theorem 4.6's title says 'three-dimensional' for four-dimensional algebras; Corollary 4.7 misstates dimension ranges (two-dimensional only has dimension 0, not 0–2); and the reference list includes several unrelated self-citations (e.g., turbine papers).

Give credit where it is due: the paper is clearly organized, it correctly recalls known classification results, and the tables are computed carefully from those classifications. But the central object is misdefined, so the main results do not stand.

Conclusion: reject. I would not send this to referees as is. If the authors want to study inner derivations, they need a definition that actually satisfies the Leibniz rule, or they should abandon the term 'inner derivation' and simply compute the full derivation algebra instead. The paper would need major restructuring and a correct central theorem before it is reviewable.

Referee Report

4 major / 4 minor

Summary. The manuscript defines inner derivations of Zinbiel algebras as maps ad_w(u) = u ∘ w − w ∘ u (Definition 3.1, with the opposite sign in Proposition 3.5), develops an algorithm for writing these maps in matrix form, and applies the algorithm to the known classifications of complex Zinbiel algebras in dimensions two, three, and four (Theorems 4.1–4.6). The central claim is Proposition 3.5, asserting that every ad_w is a derivation in the usual Leibniz-rule sense. The paper also states structural results about right and left multiplication operators and about the inner derivation space, and it concludes with a dimension table for low-dimensional cases.

Significance. If the central claim were correct, the paper would provide a useful algorithmic reference and explicit matrix descriptions of inner derivations for low-dimensional Zinbiel algebras. However, the central claim is false: the proof of Proposition 3.5 relies on an incorrect restatement of the Zinbiel identity, and the paper's own four-dimensional classification gives a direct counterexample to the Leibniz property. The tables in Theorem 4.6 consequently describe linear maps that are generally not derivations. I see no machine-checked verification or independent computation in the manuscript that could offset this error.

major comments (4)

[§3, Proposition 3.5; Theorem 4.6] The proof of Proposition 3.5 says 'Using the Zinbiel identity (u ∘ v) ∘ w = u ∘ (v ∘ w)', but Definition 2.1 defines a Zinbiel algebra by (u ∘ v) ∘ w = u ∘ (v ∘ w) + u ∘ (w ∘ v). The omitted term is essential. In the algebra A1_4 from Theorem 4.5, with e1∘e1=e2, e1∘e2=e3, e2∘e1=2e3, e1∘e3=e4, e2∘e2=3e4, e3∘e1=3e4, Definition 3.1 gives ad_{e1}(e1∘e2) = ad_{e1}(e3) = e3∘e1 − e1∘e3 = 3e4 − e4 = 2e4. The Leibniz right-hand side is ad_{e1}(e1)∘e2 + e1∘ad_{e1}(e2) = 0 + e1∘e3 = e4. Since 2e4 ≠ e4, the map ad_{e1} is not a derivation. The sign variant ad_w(u) = w∘u − u∘w in Proposition 3.5 fails in the same way up to sign, so the issue is not a sign convention. Therefore Proposition 3.5 is false, and the matrices listed in Theorem 4.6 are not matrices of derivations.
[§2, Lemma 2.6] Lemma 2.6 claims that R(A) = {R_u} and L(A) = {L_u} are subalgebras of Der(A). This is false. In the same algebra A1_4, R_{e1}(e1∘e2) = R_{e1}(e3) = e3∘e1 = 3e4, while the Leibniz rule for R_{e1} would require R_{e1}(e1)∘e2 + e1∘R_{e1}(e2) = (e1∘e1)∘e2 + e1∘(e2∘e1) = e2∘e2 + e1∘(2e3) = 3e4 + 2e4 = 5e4. Similarly, L_{e1}(e1∘e1) = L_{e1}(e2) = e3, while the Leibniz right-hand side is e2∘e1 + e1∘e2 = 2e3 + e3 = 3e3. Thus right and left multiplication operators are generally not derivations, and Lemma 2.6 is contradicted by the paper's own classification.
[§3, Lemma 3.4] The proof of Lemma 3.4 invokes the identity [w,[w',u]] = [[w,w'],u] as 'the Zinbiel identity', but this Jacobi-type identity is not a consequence of Definition 2.1 and is generally false. In A1_4, take the commutator [x,y] = x∘y − y∘x, w = e2, w' = e1, and u = e1. Then [e2,[e1,e1]] = [e2,0] = 0, whereas [[e2,e1],e1] = [e3,e1] = e3∘e1 − e1∘e3 = 3e4 − e4 = 2e4. The assertion that Inn(A) is an ideal of Der(A) is therefore unsupported; moreover, by the failure of Proposition 3.5, Inn(A) as defined is not even contained in Der(A).
[§4, Theorems 4.5–4.6 and Corollary 4.7] The classification-dependent parts of the paper are internally inconsistent. Theorem 4.6 is headed 'three-dimensional' but its table concerns the four-dimensional algebras A1_4,...,A16_4. Theorem 4.5 lists A12_4 through A16_4 with the identical multiplication rules e1∘e2=e3 and e2∘e1=e4, yet asserts that the algebras are pairwise non-isomorphic; the parameters distinguishing them are not given. The conclusion states that the dimension of inner derivations ranges between zero and two for two-dimensional algebras, but Theorem 4.2 gives dimension zero, and it attributes the zero-to-three range to three-dimensional algebras rather than four-dimensional ones, contradicting Corollary 4.7. These inconsistencies make the tables unreliable as a reference even aside from the false derivation claim.

minor comments (4)

[Definition 3.1 and Proposition 3.5] The sign convention for ad_w is inconsistent: Definition 3.1 sets ad_w(u) = u∘w − w∘u, while Proposition 3.5 and Theorem 4.2 use ad_w(u) = w∘u − u∘w. The proof of Theorem 4.4 for A4_3 computes with the latter sign but the displayed table entry for A4_3 has the opposite sign, so even the paper's own tables are not consistent with its proofs.
[§2, Proposition 2.4] Proposition 2.4(ii) has a sign error: for a derivation d, [L_u,L_d](v) = L_u(L_d(v)) − L_d(L_u(v)) = u∘d(v) − d(u∘v) = u∘d(v) − d(u)∘v − u∘d(v) = −d(u)∘v, so the bracket equals −L_{d(u)} rather than L_{d(u)} as stated.
[§3, Definition 3.2 and Lemma 3.3] Definition 3.2 writes 'A∘u = [a,u]', mixing the Zinbiel product with the commutator bracket. In Lemma 3.3, the proof does not actually verify two-sided ideal closure: for u ∈ Ann_R(A) it asserts u∘v ∈ Ann_R(A) without checking that a∘(u∘v) = 0 for all a ∈ A, and similarly for the left annihilator.
[§4, algorithm setup] The algorithm section begins 'Let A be an n-dimensional associative algebra', although the paper concerns Zinbiel algebras, and the structure constants γ^j_{it} appearing in the coefficient formula are never defined.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the quoted derivation chain is a direct multiplication-table computation; the paper's main defect is a false identity inside Proposition 3.5, which is a correctness error, not a circular reduction.

full rationale

The paper's computation of inner derivations is not circular in any of the senses defined above. Definition 3.1 introduces ad_w(u)=u∘w−w∘u and labels it an inner derivation, but the paper then attempts to prove the derivation property in Proposition 3.5. That proof is invalid because it replaces the Zinbiel identity (u∘v)∘w = u∘(v∘w)+u∘(w∘v) with the associative-looking identity (u∘v)∘w = u∘(v∘w), omitting the second term. This is a mathematical error—the claimed maps are not derivations on the paper's own algebra A^1_4—but it is not a case of a prediction reducing to a fitted input, a self-citation carrying the load, or a definition that already contains the target result. The matrices in Theorem 4.6 are obtained by direct computation from the externally cited classification of Zinbiel algebras; they do not derive their content from any fitted parameter or from the authors' prior work. The heavy presence of self-citations in the bibliography is not load-bearing for the main derivation chain, which uses external classification results [12,13,14,15,16]. Therefore the circularity score is 0, with the caveat that correctness concerns about Proposition 3.5 should be assessed separately.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on the Zinbiel identity, the completeness of the cited classifications of low-dimensional Zinbiel algebras, and an unjustified assumption that the commutator map ad_w is automatically a derivation. There are no fitted parameters or new entities. The invalid commutator assumption is not a circular derivation from self-authored results; it is simply false.

assumptions (3)

standard math Zinbiel identity: (u∘v)∘w = u∘(v∘w)+u∘(w∘v) for all u,v,w.
Defining property invoked throughout the paper, stated in Definition 2.1.
domain assumption The classifications of 2-, 3-, and 4-dimensional complex Zinbiel algebras cited from [12-16] are complete and correct.
Theorems 4.1, 4.3, and 4.5 rely on these lists without independent verification.
ad hoc to paper The commutator map ad_w satisfies the Leibniz rule for Zinbiel algebras.
Assumed in Proposition 3.5 using a false identity; contradicted by the paper's own A1_4 table.

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Pith. "Pith review of An Algorithmic Approach to Inner Derivations of Low-Dimensional Zinbiel Algebras." pith.science (2026). https://pith.science/paper/XIHEGE45

@misc{pith2026241220599,
  author       = {Pith},
  title        = {Pith review of: An Algorithmic Approach to Inner Derivations of Low-Dimensional Zinbiel Algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XIHEGE45}},
  note         = {Machine review of arXiv:2412.20599}
}

read the original abstract

In this paper, we introduce the concept of inner derivations of low-dimensional Zinbiel algebras and investigate their properties. The primary objective of this study is to develop an algorithm to characterize the inner derivations of any n-dimensional Zinbiel algebra in matrix form. Additionally, we apply this algorithm to two, three and four-dimensional complex Zinbiel algebras, providing explicit descriptions of their inner derivations.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Computational Approaches to Derivations and Automorphism Groups of Associative Algebras
math.RA 2025-01 reject novelty 4.0 of 10

The paper claims to compute derivations and automorphism groups of low-dimensional associative algebras over C, but the computations are not self-contained and contain errors.

Reference graph

Works this paper leans on

37 extracted references · 15 canonical work pages · cited by 1 Pith paper

[1]

Loday, J. L. (1995). Cup-product for Leibniz cohomology a nd dual Leibniz algebras. Mathematica Scandinavica, 189-1 96

work page 1995
[2]

Loday, J. L. (1993). Une version non commutative des algeb res de Lie: les algebres de Leibniz. Les rencontres physicie ns- math´ ematiciens de Strasbourg-RCP25, 44, 127-151

work page 1993
[3]

Dokas, I. (2010). Zinbiel algebras and commutative algeb ras with divided powers. Glasgow Mathematical Journal, 52( 2), 303-313

work page 2010
[4]

Ayala, V., Kizil, E., & de Azevedo Tribuzy, I. (2012). On an algorithm for ﬁnding derivations of Lie algebras. Proyecci ones (Antofagasta), 31(1), 81-90

work page 2012
[5]

Xiao, Z., & Wei, F. (2012). Lie triple derivations of trian gular algebras. Linear algebra and its applications, 437(5 ), 1234-1249

work page 2012
[6]

Zhou, J. (2013). Triple derivations of perfect Lie algebr as. Communications in Algebra, 41(5), 1647-1654

work page 2013
[7]

A., Mohammed, N

Fiidow, M. A., Mohammed, N. F., Rakhimov, I. S., Husain, S. K. S., Husain, S. K. S., & Basri, W. (2010). On inner derivations of ﬁnite dimensional associative algebras. to appear

work page 2010
[8]

Patlertsin, S., Pongprasert, S., & Rungratgasame, T. (20 24). On Inner Derivations of Leibniz Algebras. Mathematics , 12(8), 1152

work page

Show all 37 references

[9]

Gao, J., Ni, J., & Yu, J. (2024). Some Results on Zinbiel Alg ebras and Rota–Baxter Operators. Axioms, 13(5), 314

2024
[12]

Almutairi, H., & AbdGhafur, A. (2018). Derivations of so me classes of Zinbiel algebras. International Journal of Pu re and Applied Mathematics, 2, 12-13

2018
[13]

Omirov, B. A. (2002). Classiﬁcation of two-dimensional complex Zinbiel algebras. Uzbek. Mat. Zh, 2, 55-59

2002
[14]

A., J´ unior, R

Alvarez, M. A., J´ unior, R. F., & Kaygorodov, I. (2022). T he algebraic and geometric classiﬁcation of Zinbiel algebr as. Journal of Pure and Applied Algebra, 226(11), 107106

2022
[15]

Ni, J. (2014). Centroids of Zinbiel algebras. Communica tions in Algebra, 42(4), 1844-1853

2014
[16]

Q., Khudoyberdiyev, A

Adashev, J. Q., Khudoyberdiyev, A. K., & Omirov, B. A. (20 10). Classiﬁcations of some classes of Zinbiel algebras. Jo urnal of Generalized Lie Theory and Applications, 4, 1-10
[17]

Sania, A., Imed, B., Mosbahi, B., & Saber, N. (2023). Coho mology of compatible BiHom-Lie algebras. arXiv preprint arXiv:2303.12906

2023 arXiv
[19]

Zahari, A., Mosbahi, B., & Basdouri, I. (2023). Classiﬁc ation, Derivations and Centroids of Low-Dimensional Compl ex BiHom-Trialgebras. arXiv preprint arXiv:2304.06781

2023 arXiv
[20]

Mosbahi, B., Zahari, A., & Basdouri, I. (2023). Classiﬁc ation, α -Inner Derivations and α -Centroids of Finite-Dimensional Complex Hom-Trialgebras. arXiv preprint arXiv:2305.0047 1

2023
[21]

Mosbahi, B., Asif, S., & Zahari, A. (2023). Classiﬁcatio n of tridendriform algebra and related structures. arXiv pr eprint arXiv:2305.08513

2023 arXiv
[22]

A., Zahari, A., & Mosbahi, B

Fiidow, M. A., Zahari, A., & Mosbahi, B. (2023). Quasi-Ce ntroids and Quasi-Derivations of Low Dimensional Associat ive Algebras. arXiv preprint arXiv:2306.14331

2023 arXiv
[23]

Asif, S., Wang, Y., Mosbahi, B., & Basdouri, I. (2023). Co homology and deformation theory of O-operators on Hom-Lie conformal algebras. arXiv preprint arXiv:2312.04121

2023 arXiv
[24]

Mansuroglu, N., & Mosbahi, B. (2024). On structures of Bi Hom-Superdialgebras and their derivations. arXiv preprin t arXiv:2404.12098

2024 arXiv
[25]

Mansuroglu, N., & Mosbahi, B. (2024). Generalized deriv ations of BiHom-supertrialgebras. arXiv preprint arXiv:2404.12112

2024 arXiv
[26]

Mainellis, E., Mosbahi, B., & Zahari, A. (2024). Cohomol ogy of BiHom-Associative Trialgebras. arXiv preprint arXiv:2404.15567

2024 arXiv
[27]

Mainellis, E., Mosbahi, B., & Zahari, A. (2024). Compati ble Associative Algebras and Some Invariants. arXiv prepri nt arXiv:2405.18243

2024 arXiv
[28]

Imed, B., & Mosbahi, B. (2024). Classiﬁcation of ( ρ, τ, σ )-derivations of two-dimensional left-symmetric dialgeb ras. arXiv preprint arXiv:2411.05716

2024 arXiv
[29]

Imed, B., Lerbet, J., & Mosbahi, B. (2024). Quasi-Centro ids and Quasi-Derivations of low-dimensional Zinbiel alge bras. arXiv preprint arXiv:2411.09532

2024 arXiv
[30]

Mosbahi, M., Elgasri, S., Lajnef, M., Mosbahi, B., & Dris s, Z. (2021). Performance enhancement of a twisted Savonius hydrokinetic turbine with an upstream deﬂector. Internati onal Journal of Green Energy, 18(1), 51-65. 16 BOUZID MOSBAHI 1, AHMED ZAHARI 2

2021
[31]

Mosbahi, M., Lajnef, M., Derbel, M., Mosbahi, B., Aric` o , C., Sinagra, M., & Driss, Z. (2021). Performance improveme nt of a drag hydrokinetic turbine. Water, 13(3), 273

2021
[32]

Mosbahi, M., Derbel, M., Lajnef, M., Mosbahi, B., Driss, Z., Aric` o, C., & Tucciarelli, T. (2021). Performance study of twisted Darrieus hydrokinetic turbine with novel blade des ign. Journal of Energy Resources Technology, 143(9), 09130 2

2021
[33]

Mosbahi, M., Lajnef, M., Derbel, M., Mosbahi, B., Driss, Z., Aric` o, C., & Tucciarelli, T. (2021). Performance impro vement of a Savonius water rotor with novel blade shapes. Ocean Engi neering, 237, 109611

2021
[34]

Mosbahi, M., Derbel, M., Hannachi, M., Mosbahi, B., Dris s, Z., Aric` o, C., & Tucciarelli, T. (2023). Performance stu dy of spiral Darrieus water rotor with V-shaped blades. Proceedi ngs of the Institution of Mechanical Engineers, Part C: Jour nal of Mechanical Engineering Sc...

2023
[35]

Mosbahi, B., Zahari, A., Basdouri, I. (2023). Classiﬁca tion, α -Inner Derivations and α -Centroids of Finite-Dimensional Complex Hom-Trialgebras. Pure and App lied Mathematics Journal, 12(5), 86-97. https://doi.org/10.11648/j.pamj.20231205.12

2023 doi
[36]

Z., & MOSBAHI, B

ABDOU, A. Z., & MOSBAHI, B. (2024). CLASSIFICATION OF COM PATIBLE ASSOCIATIVE ALGEBRAS AND SOME INV ARIANTS. Available at SSRN 4877916

2024
[37]

Makhlouf, A., & Zahari, A. (2020). Structure and classiﬁ cation of Hom-associative algebras. Acta et Commentatione s Universitatis Tartuensis de Mathematica, 24(1), 79-102

2020
[38]

On BiHom-Associative dialgebr as.Open Journal of Mathematical Sciences, Vol

Zahari, A.; Bakayoko, I. On BiHom-Associative dialgebr as.Open Journal of Mathematical Sciences, Vol. 7, No. 1 (202 3).pp. 96-117
[39]

Imed, B., Lerbet, J., & Mosbahi, B. (2024). Central deriv ations of low-dimensional Zinbiel algebras. arXiv preprin t arXiv:2411.15642

2024 arXiv
[40]

Okba, B., & Mosbahi, B. (2024). Rota-type operators on 2- dimensional dendriform algebras. arXiv preprint arXiv:2411.15358. Department of Mathematics, F aculty of Sciences, Universit y of Sfax, Sfax, Tunisia 2 IRIMAS-Department of Mathematics, F aculty of Sciences, University ...

2024 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Loday, J. L. (1995). Cup-product for Leibniz cohomology a nd dual Leibniz algebras. Mathematica Scandinavica, 189-1 96

work page 1995

[2] [2]

Loday, J. L. (1993). Une version non commutative des algeb res de Lie: les algebres de Leibniz. Les rencontres physicie ns- math´ ematiciens de Strasbourg-RCP25, 44, 127-151

work page 1993

[3] [3]

Dokas, I. (2010). Zinbiel algebras and commutative algeb ras with divided powers. Glasgow Mathematical Journal, 52( 2), 303-313

work page 2010

[4] [4]

Ayala, V., Kizil, E., & de Azevedo Tribuzy, I. (2012). On an algorithm for ﬁnding derivations of Lie algebras. Proyecci ones (Antofagasta), 31(1), 81-90

work page 2012

[5] [5]

Xiao, Z., & Wei, F. (2012). Lie triple derivations of trian gular algebras. Linear algebra and its applications, 437(5 ), 1234-1249

work page 2012

[6] [6]

Zhou, J. (2013). Triple derivations of perfect Lie algebr as. Communications in Algebra, 41(5), 1647-1654

work page 2013

[7] [7]

A., Mohammed, N

Fiidow, M. A., Mohammed, N. F., Rakhimov, I. S., Husain, S. K. S., Husain, S. K. S., & Basri, W. (2010). On inner derivations of ﬁnite dimensional associative algebras. to appear

work page 2010

[8] [8]

Patlertsin, S., Pongprasert, S., & Rungratgasame, T. (20 24). On Inner Derivations of Leibniz Algebras. Mathematics , 12(8), 1152

work page