The relationship between local derivations and local automorphisms of some associative algebras

Abduqaxxor Qurbonov; Farkhodzhon Arzikulov; Utkir Khakimov

arxiv: 2511.17316 · v2 · submitted 2025-11-21 · 🧮 math.RA

The relationship between local derivations and local automorphisms of some associative algebras

Farkhodzhon Arzikulov , Utkir Khakimov , Abduqaxxor Qurbonov This is my paper

Pith reviewed 2026-05-17 06:55 UTC · model grok-4.3

classification 🧮 math.RA

keywords local derivationslocal automorphismsnilpotent associative algebrasnaturally graded algebrasmatrix representationsLie algebra structurefive-dimensional algebras

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The pith

Two five-dimensional nilpotent associative algebras have local derivations and automorphisms that are not global ones, and their local derivations form Lie algebras.

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

The paper investigates local derivations and local automorphisms on the five-dimensional naturally graded nilpotent associative algebras π₂ and π₃. It determines the general forms of the matrices for these local maps and shows that they are more general than the matrices for actual automorphisms and derivations. This establishes the existence of local maps that fail to be global on these algebras. The paper also shows a connection between local automorphisms and local derivations using an exponential expression and proves that the local derivations form a Lie algebra under the Lie bracket operation.

Core claim

The general form of the matrix of an automorphism or derivation on these algebras does not coincide with that of a local automorphism or local derivation. Therefore these algebras admit local automorphisms and local derivations that are not automorphisms or derivations. A relationship between local automorphisms and local derivations is established via an exponential expression. The sets of local derivations on π₂ and π₃ form Lie algebras with respect to the Lie brackets, providing a positive solution to the Lie algebra problem for local derivations on these algebras.

What carries the argument

The general matrix forms of maps with respect to the fixed basis of the algebras π₂ and π₃, which distinguish local from global maps.

If this is right

The algebras π₂ and π₃ possess local automorphisms that are not automorphisms.
The algebras π₂ and π₃ possess local derivations that are not derivations.
The sets of local derivations of π₂ and π₃ form Lie algebras.
Local automorphisms and local derivations are related by an exponential expression.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar distinctions between local and global maps may appear in other low-dimensional nilpotent algebras with the same grading.
Matrix computations of this type could be used to check the Lie algebra property for local derivations in higher dimensions.
The exponential relation might indicate a way to generate local maps from derivations in broader classes of algebras.

Load-bearing premise

The algebras π₂ and π₃ are exactly the five-dimensional naturally graded nilpotent associative algebras with the specific multiplication tables and basis vectors fixed in the paper.

What would settle it

A direct computation showing that every local derivation on π₂ or π₃ is actually a derivation would falsify the existence of strictly local derivations.

read the original abstract

In the present paper, local derivations and local automorphisms of five-dimensional naturally graded nilpotent associative algebras are studied. Namely, a general form of the matrices of local derivations and local automorphisms of algebras $\pi_2$ and $\pi_3$ is clarified. It turns out that the general form of the matrix of an automorphism (derivation) on these algebras does not coincide with the local automorphism's (resp. local derivation's) matrix's general form on these algebras. Therefore, these associative algebras have local automorphisms (resp. local derivations) that are not automorphisms (resp. derivations). We also establish a relationship between local automorphisms and local derivations via an exponential expression. We prove that the sets of local derivations of algebras $\pi_2$ and $\pi_3$ form Lie algebras with respect to the Lie brackets. Thus, we show that the Lie algebra problem from the Ayupov-Eldique-Kudaybergenov problems for local derivations of the algebras under consideration has a positive solution. The remaining problems from the Ayupov-Eldique-Kudaybergenov problems also have a positive solution for algebras $\pi_2$ and $\pi_3$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Explicit matrix calculations for π2 and π3 show local derivations and automorphisms properly contain the global ones and that the local derivations close under Lie brackets.

read the letter

Hi, the core of this paper is a set of explicit matrix descriptions for local derivations and local automorphisms on the two fixed five-dimensional nilpotent algebras π2 and π3. The authors derive the general forms by imposing the local condition on each basis vector, solve the resulting linear systems from the structure constants, and find that these forms have extra parameters compared with the global derivation and automorphism matrices. They also record an exponential relation linking the local versions and verify that the local derivations are closed under the commutator bracket, which gives a positive answer to the Lie-algebra part of the Ayupov-Eldique-Kudaybergenov questions for these algebras. The remaining problems in that list are stated to have positive solutions here as well. What is actually new is the concrete parameter counts and matrix entries for these two specific multiplication tables; the calculations appear to be direct extensions of earlier work on local maps but supply the missing details for π2 and π3. The approach is straightforward and appropriate for low-dimensional graded algebras: fix the basis, write the local condition as a family of derivation or automorphism equations, and compare dimensions. That part is done cleanly enough to be useful as a data point. The soft spots are modest but real. Everything rests on the accuracy of translating the algebra products into the linear conditions on the matrix entries; a slip in one of those equations or in the basis ordering would change the reported parameter counts and therefore the claims of proper containment and Lie closure. The paper stays within these two examples and does not attempt broader theorems, so its reach is narrow. No free parameters or invented objects appear, and the work is not circular. This is for readers already following the literature on local derivations in associative algebras and the specific open questions cited. Someone outside that niche will not find general methods or applications. A serious referee should see it. The computations are finite and checkable, and if the matrix arithmetic holds, the results add verifiable instances to the record without overclaiming.

Referee Report

0 major / 4 minor

Summary. The paper examines local derivations and local automorphisms on the five-dimensional naturally graded nilpotent associative algebras π₂ and π₃. It derives the general matrix forms for these local maps, demonstrates that they do not coincide with the forms for global derivations and automorphisms, establishes an exponential relationship between local automorphisms and local derivations, and proves that the local derivations of these algebras form Lie algebras with respect to the commutator bracket. Consequently, it provides positive solutions to the Lie algebra problem and other problems posed by Ayupov, Elduque, and Kudaybergenov for these algebras.

Significance. Assuming the explicit algebraic computations with the structure constants are correct, this manuscript offers concrete examples of associative algebras where local derivations and automorphisms properly extend the global ones. The confirmation that local derivations form a Lie algebra addresses a specific open question in the literature on local maps, providing verifiable instances in low-dimensional nilpotent algebras. The direct matrix approach allows for reproducibility of the results.

minor comments (4)

The abstract states that the remaining problems from the Ayupov-Eldique-Kudaybergenov list also have positive solutions, but the body of the paper does not enumerate which specific problems beyond the Lie-algebra closure are addressed; adding a short list in the introduction or conclusion would improve clarity.
In Section 2, the multiplication tables for π₂ and π₃ are given but omit an explicit listing of all nonzero structure constants; including the full set of products (e.g., e₁e₂ = e₃ for π₂) would facilitate independent verification of the linear systems solved later.
The field of scalars is not stated explicitly in the matrix computations (presumably ℂ or an algebraically closed field of characteristic zero); adding this once in the preliminaries would remove ambiguity.
The exponential relation in Section 4 is presented via matrix exponentiation, but a brief remark on why the nilpotency ensures the series terminates after finitely many terms would aid readers unfamiliar with the finite-dimensional setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for recommending minor revision. The referee's summary correctly reflects our main results on the matrix forms of local derivations and local automorphisms for the algebras π₂ and π₃, their distinction from global maps, the exponential relationship, and the resolution of the relevant Ayupov-Eldique-Kudaybergenov problems.

Circularity Check

0 steps flagged

No circularity: explicit matrix computations on fixed structure constants

full rationale

The paper derives the general matrix forms of local derivations and local automorphisms for the concrete algebras π₂ and π₃ by imposing the local condition (for each basis element e_i there exists a global derivation/automorphism B_i with D(e_i)=B_i(e_i)) and solving the resulting linear systems over the given multiplication tables. The distinction from global maps and the Lie-algebra closure under commutators follow directly from the dimension and parameter counts of those solution spaces. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear; the results are self-contained algebraic calculations independent of prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the concrete multiplication rules of the two named algebras and on the standard definitions of local derivation and local automorphism; no free parameters or new entities are introduced.

axioms (2)

domain assumption The algebras π2 and π3 are five-dimensional naturally graded nilpotent associative algebras with fixed basis and structure constants as defined in the paper.
All matrix computations and closure properties depend on these specific structure constants.
standard math Local derivations and local automorphisms are defined in the usual way for associative algebras (a map D is a local derivation if for every x there exists a derivation D_x such that D(x) = D_x(x)).
The paper works within the established framework of local maps in ring theory.

pith-pipeline@v0.9.0 · 5527 in / 1624 out tokens · 18814 ms · 2026-05-17T06:55:53.720578+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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Journal of Geometry and Physics 178: 104573 DOI: 10.1016/j.geomphys.2022.104573 22 REFERENCES

Arzikulov FN, Karimjanov IA, Umrzaqov SM (2022) Local and 2-local automorphisms of some solvable Leibniz algebras. Journal of Geometry and Physics 178: 104573 DOI: 10.1016/j.geomphys.2022.104573 22 REFERENCES

work page doi:10.1016/j.geomphys.2022.104573 2022
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Scientific Bulletin of Namangan State University No 7: 38–42

Arzikulov FN, Qurbonov A (2025) On automorphisms of finite- dimensional naturally graded associative algebras. Scientific Bulletin of Namangan State University No 7: 38–42

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Algebra, Complex Anal- ysis and Pluripotential Theory, USUZCAMP

Ayupov ShA, Kudaybergenov KK (2017) Local Automorphisms on Finite-Dimensional Lie and Leibniz Algebras. Algebra, Complex Anal- ysis and Pluripotential Theory, USUZCAMP. Springer Proceedings in Mathematics and Statistics 264: 31–44. DOI: 10.1007/978-3-030-01144-43

work page doi:10.1007/978-3-030-01144-43 2017
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Bulletin of the Malaysian Mathematical Sciences Society 43: 2199–2234

Ayupov ShA, Kudaybergenov KK, Omirov BA (2020) Local and 2- local derivations and automorphisms on simple Leibniz algebras. Bulletin of the Malaysian Mathematical Sciences Society 43: 2199–2234. DOI: 10.1007/s40840-019-00799-5

work page doi:10.1007/s40840-019-00799-5 2020
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Journal of Pure and Applied Algebra 227(5): 107277

Ayupov ShA, Elduque A, Kudaybergenov KK (2023) Local derivations and automorphisms of Cayley algebras. Journal of Pure and Applied Algebra 227(5): 107277. DOI: 10.1016/j.jpaa.2022.107277

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Uzbek Mathematical Journal No 2: 27–34

Becker T, Escobar J, Salas C, Turdibaev R (2019) On Local Auto- morphisms of sl2. Uzbek Mathematical Journal No 2: 27–34. DOI: 10.29229/uzmj.2019-2-3

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Bulletin of Irkutsk State University

Elisova AP, Zotov IN, Levchuk VM, Suleimanova GS (2011) Local automorphisms and local derivations of nilpotent matrix algebras. Bulletin of Irkutsk State University. Series Mathematics 4(1): 9–19. https://www.mathnet.ru/iigum90

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Johnson B (2001) Local derivations on C ∗-algebras are derivations. Trans. Amer. Math. Soc. 353: 313–325. DOI: 10.2307/221975

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Liu J-H, Wong N-Ch (2007) Local automorphisms of operator algebras. Taiwanese Journal of Mathematics 11(3): 611–619. DOI: 10.11650/twjm/1500404747

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DOI: 10.1007/s10114-018-7519-6

work page doi:10.1007/s10114-018-7519-6

[1] [1]

Journal of Geometry and Physics 178: 104573 DOI: 10.1016/j.geomphys.2022.104573 22 REFERENCES

Arzikulov FN, Karimjanov IA, Umrzaqov SM (2022) Local and 2-local automorphisms of some solvable Leibniz algebras. Journal of Geometry and Physics 178: 104573 DOI: 10.1016/j.geomphys.2022.104573 22 REFERENCES

work page doi:10.1016/j.geomphys.2022.104573 2022

[2] [2]

Scientific Bulletin of Namangan State University No 7: 38–42

Arzikulov FN, Qurbonov A (2025) On automorphisms of finite- dimensional naturally graded associative algebras. Scientific Bulletin of Namangan State University No 7: 38–42

work page 2025

[3] [3]

Algebra, Complex Anal- ysis and Pluripotential Theory, USUZCAMP

Ayupov ShA, Kudaybergenov KK (2017) Local Automorphisms on Finite-Dimensional Lie and Leibniz Algebras. Algebra, Complex Anal- ysis and Pluripotential Theory, USUZCAMP. Springer Proceedings in Mathematics and Statistics 264: 31–44. DOI: 10.1007/978-3-030-01144-43

work page doi:10.1007/978-3-030-01144-43 2017

[4] [4]

Bulletin of the Malaysian Mathematical Sciences Society 43: 2199–2234

Ayupov ShA, Kudaybergenov KK, Omirov BA (2020) Local and 2- local derivations and automorphisms on simple Leibniz algebras. Bulletin of the Malaysian Mathematical Sciences Society 43: 2199–2234. DOI: 10.1007/s40840-019-00799-5

work page doi:10.1007/s40840-019-00799-5 2020

[5] [5]

Journal of Pure and Applied Algebra 227(5): 107277

Ayupov ShA, Elduque A, Kudaybergenov KK (2023) Local derivations and automorphisms of Cayley algebras. Journal of Pure and Applied Algebra 227(5): 107277. DOI: 10.1016/j.jpaa.2022.107277

work page doi:10.1016/j.jpaa.2022.107277 2023

[6] [6]

Uzbek Mathematical Journal No 2: 27–34

Becker T, Escobar J, Salas C, Turdibaev R (2019) On Local Auto- morphisms of sl2. Uzbek Mathematical Journal No 2: 27–34. DOI: 10.29229/uzmj.2019-2-3

work page doi:10.29229/uzmj.2019-2-3 2019

[7] [7]

Linear Algebra Appl

Costantini M (2019) Local automorphisms of finite dimensional simple Lie algebras. Linear Algebra Appl. 562: 123–134. DOI: 10.1016/j.laa.2018.10.009

work page doi:10.1016/j.laa.2018.10.009 2019

[8] [8]

Bulletin of Irkutsk State University

Elisova AP, Zotov IN, Levchuk VM, Suleimanova GS (2011) Local automorphisms and local derivations of nilpotent matrix algebras. Bulletin of Irkutsk State University. Series Mathematics 4(1): 9–19. https://www.mathnet.ru/iigum90

work page 2011

[9] [9]

Johnson B (2001) Local derivations on C ∗-algebras are derivations. Trans. Amer. Math. Soc. 353: 313–325. DOI: 10.2307/221975

work page doi:10.2307/221975 2001

[10] [10]

Kadison R (1990) Local derivations. J. Algebra 130: 494–509. DOI: 10.1016/0021-8693(90)90095-6

work page doi:10.1016/0021-8693(90)90095-6 1990

[11] [11]

Mediterr

Karimjanov IA, Ladra M (2020) Some classes of nilpotent associative algebras. Mediterr. J. Math. 17(2): 1–21. https://doi.org/10.1007/s00009- 020-1504-x

work page doi:10.1007/s00009- 2020

[12] [12]

Taiwanese Journal of Mathematics 11(3): 611–619

Liu J-H, Wong N-Ch (2007) Local automorphisms of operator algebras. Taiwanese Journal of Mathematics 11(3): 611–619. DOI: 10.11650/twjm/1500404747

work page doi:10.11650/twjm/1500404747 2007

[13] [13]

Acta Mathematica Sinica, English Series 35(3): 427–

Wang Y, Chen H, Nan J (2019) 2-Local automorphisms on basic classical Lie superalgebras. Acta Mathematica Sinica, English Series 35(3): 427–

work page 2019

[14] [14]

DOI: 10.1007/s10114-018-7519-6

work page doi:10.1007/s10114-018-7519-6