pith. sign in

arxiv: 2605.19512 · v1 · pith:CC45RRWInew · submitted 2026-05-19 · 🧮 math.RA · math.GR

Images of Lie Polynomials on simple Lie algebras

Harish Kishnani , Anupam Singh This is my paper

Pith reviewed 2026-05-20 02:06 UTC · model grok-4.3

classification 🧮 math.RA math.GR
keywords Lie polynomialsimages of mapssimple Lie algebrasChevalley algebrasfinite fieldsautomorphismsconjugacy classes
0
0 comments X

The pith

For simple Chevalley algebras over finite fields of very good characteristic, any automorphism-closed subset containing zero arises as the image of a Lie polynomial.

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

The paper establishes a classification for the possible images of Lie polynomials evaluated on simple Chevalley Lie algebras defined over finite fields in very good characteristic. It shows that the only conditions needed to guarantee a subset is such an image are that the subset is invariant under all automorphisms of the algebra and contains the zero element. This mirrors earlier results on word maps for finite simple groups and provides a complete description of attainable image sets without first constructing explicit polynomials. The authors also give concrete Lie polynomials realizing each conjugacy class union zero inside the special linear Lie algebra of rank one over odd finite fields.

Core claim

We prove that for a simple Chevalley algebra over a finite field of very good characteristic, these two properties are enough to classify all possible subsets that can be the image of a Lie polynomial.

What carries the argument

The pair of conditions (automorphism invariance and containment of zero) that together characterize every attainable image set of a Lie polynomial.

If this is right

  • Every GL_2(q)-conjugacy class together with zero arises as the image of an explicit Lie polynomial when the algebra is sl_2(q) for q odd.
  • The classification applies uniformly to all simple Chevalley algebras meeting the characteristic and finiteness hypotheses.
  • Finding explicit Lie polynomials for a prescribed admissible subset becomes the remaining constructive task after the classification is settled.

Where Pith is reading between the lines

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

  • The result indicates that the finite-field setting permits a coarser description of images than the algebraically closed case, where additional restrictions appear.
  • Computational checks for small-rank algebras and small fields could verify the classification by enumerating all possible Lie polynomials of low degree.
  • The parallel with Lubotzky's theorem on word maps suggests that similar image problems for other non-associative structures over finite fields may admit analogous two-condition classifications.

Load-bearing premise

The Lie algebra must be a simple Chevalley algebra over a finite field of very good characteristic.

What would settle it

A concrete subset of such an algebra that is closed under all automorphisms and contains zero yet cannot be obtained as the image of any Lie polynomial, or an image that fails one of the two properties.

read the original abstract

A Lie polynomial is an element of a free Lie algebra $\mathcal F_k$ on $k$-generators, which defines a Lie map on a given Lie algebra $L$, by substituting $k$-elements of $L$. Similar to word maps on groups and polynomial maps on algebras, one studies here questions analogous to Waring-like problems, the L'vov-Kaplansky conjecture, etc. In this article, we would like to address a problem for Lie algebras parallel to the one Lubotzky solved (Images of word maps in finite simple groups, Glasg. Math. J., 56, no. 2, 465-469, 2014) for finite simple groups. It is easy to verify that the image of a Lie map is (a) closed under automorphism, and (b) contains $0$. In this article, we prove that for a simple Chevalley algebra over a finite field of ``very good'' characteristic, these two properties are enough to classify all possible subsets that can be the image of a Lie polynomial. The next question is to find such Lie polynomials for a given subset satisfying the two properties. Contrary to the results over an algebraically closed field, we find Lie polynomials in the case of Lie algebra $\mathfrak{sl}_2(q)$, for $q$ odd, which give each $\rm{GL}_2(q)$ conjugacy class together with zero as an image.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript proves that for a simple Chevalley Lie algebra L over a finite field of very good characteristic, the images of Lie polynomials are precisely the subsets of L that are invariant under Aut(L) and contain 0. The 'only if' direction follows directly from the definitions; the converse is established by showing existence, with explicit constructions provided for the case of sl_2(q) (q odd) realizing each GL_2(q)-conjugacy class union {0}.

Significance. If the result holds, this gives a complete classification of images of Lie polynomials on these algebras, directly paralleling Lubotzky's theorem on word maps in finite simple groups and advancing the study of Waring-type and L'vov-Kaplansky problems in the Lie-algebra setting. The explicit constructions for sl_2(q) are a concrete strength, as they realize images not known to exist over algebraically closed fields.

major comments (1)
  1. [Abstract (paragraph beginning 'In this article, we prove that...')] Abstract, paragraph beginning 'In this article, we prove that...': the central classification claim requires proving existence of a Lie polynomial for every Aut(L)-invariant subset containing 0. Explicit constructions are supplied only for sl_2(q) with q odd; the argument for higher-rank and exceptional types appears to rest on an unstated reduction or non-constructive step whose details are not verified in the manuscript, which is load-bearing for the general statement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying the need for additional clarity on the existence direction of the classification. We address the concern point by point below.

read point-by-point responses

  1. Referee: Abstract, paragraph beginning 'In this article, we prove that...': the central classification claim requires proving existence of a Lie polynomial for every Aut(L)-invariant subset containing 0. Explicit constructions are supplied only for sl_2(q) with q odd; the argument for higher-rank and exceptional types appears to rest on an unstated reduction or non-constructive step whose details are not verified in the manuscript, which is load-bearing for the general statement.

    Authors: We agree that the current exposition does not sufficiently detail the reduction used to establish existence in higher-rank and exceptional cases. The 'only if' direction is immediate from the definitions. For the converse, the argument proceeds by reducing to the sl_2(q) case via the existence of suitable sl_2-subalgebras whose Aut(L)-orbits generate the desired invariant sets, combined with an extension of the explicit sl_2 polynomials; however, this reduction is only sketched and not verified in full for all types. We will add a new subsection that makes the reduction explicit, states the necessary embedding lemmas, and confirms applicability to exceptional types. revision: yes

Circularity Check

0 steps flagged

No circularity: classification follows from definitions plus explicit constructions

full rationale

The paper states that images of Lie polynomials are always Aut-invariant and contain 0 (immediate from the definition of Lie maps and the action of automorphisms). The converse—that every such subset arises as an image—is asserted as a theorem for simple Chevalley algebras over finite fields of very good characteristic, with explicit Lie polynomials constructed for each GL_2(q)-conjugacy class union {0} when L = sl_2(q), q odd. No equations, fitted parameters, or self-citations are shown to reduce the existence claim to a tautology or to prior work by the same authors; the derivation therefore remains non-circular and self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the domain assumption that the Lie algebra is simple Chevalley of very good characteristic; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption The Lie algebra is a simple Chevalley algebra over a finite field of very good characteristic.
    Explicitly required in the statement of the classification theorem.

pith-pipeline@v0.9.0 · 5788 in / 1112 out tokens · 42397 ms · 2026-05-20T02:06:13.186370+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    GAP -- Groups, Algorithms, and Programming, Version 4.14.0. 2026

    work page 2026

  2. [2]

    Mathematika , VOLUME =

    Panja, Saikat and Saini, Prachi and Singh, Anupam , TITLE =. Mathematika , VOLUME =. 2025 , Number =

    work page 2025

  3. [3]

    Panja, Saikat and Saini, Prachi and Singh, Anupam , TITLE =. Comm. Algebra , FJOURNAL =. 2026 , Number =

    work page 2026

  4. [4]

    Israel J

    Kishore, Krishna and Singh, Anupam , TITLE =. Israel J. Math. , FJOURNAL =. 2025 , NUMBER =

    work page 2025

  5. [5]

    Larsen, Michael and Shalev, Aner and Tiep, Pham Huu , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2011 , NUMBER =

    work page 2011

  6. [6]

    and Liebeck, Martin W

    Guralnick, Robert M. and Liebeck, Martin W. and O'Brien, E. A. and Shalev, Aner and Tiep, Pham Huu , title=. Invent. Math. , volume=. 2018 , number=

    work page 2018

  7. [7]

    Gordeev, N. L. and Kunyavski. Word maps and word maps with constants of simple algebraic groups , JOURNAL =. 2016 , Number =

    work page 2016

  8. [8]

    Kanel-Belov, Alexey and Malev, Sergey and Rowen, Louis , TITLE =. Proc. Amer. Math. Soc. , VOLUME =. 2016 , NUMBER =

    work page 2016

  9. [9]

    Bre. The. Bull. Lond. Math. Soc. , VOLUME =. 2023 , NUMBER =

    work page 2023

  10. [10]

    The image of

    Centrone, Lucio and F. The image of. J. Algebra , FJOURNAL =. 2024 , PAGES =

    work page 2024

  11. [11]

    Kanel-Belov, Alexei and Malev, Sergey and Rowen, Louis , TITLE =. Comm. Algebra , FJOURNAL =. 2017 , NUMBER =

    work page 2017

  12. [12]

    Linear Algebra Appl

    Gargate, Ivan Gonzales and Mello, Thiago Castilho de , TITLE =. Linear Algebra Appl. , VOLUME =. 2023 , PAGES =

    work page 2023

  13. [13]

    Israel J

    Gargate, Ivan Gonzales and Mello, Thiago Castilho de , TITLE =. Israel J. Math. , VOLUME =. 2022 , NUMBER =

    work page 2022

  14. [14]

    Akhiezer, Dmitri , Title =. Mosc. Math. J. , Volume=. 2015 , number =

    work page 2015

  15. [15]

    Brown, Gordon , titl=. Proc. Amer. Math. Soc. , volume=. 1963 , page=

    work page 1963

  16. [16]

    Kramer, Linus , title=. Innov. Incidence Geom. , volume=. 2023 , number=

    work page 2023

  17. [17]

    Lubotzky, Alexander , title=. Glasg. Math. J. , volume=

    work page

  18. [18]

    Chuang, Chen-Lian , title=. Proc. Amer. Math. Soc. , volume=. 1990 , number=

    work page 1990

  19. [19]

    and Emrich, Zachary M

    Anzis, Benjamin E. and Emrich, Zachary M. and Valiveti, Kaavya G. , title=. Linear Algebra Appl. , volume=. 2015 , pages=

    work page 2015

  20. [20]

    Nehra, Niranjan and Rani, Shushma , title=. Comm. Algebra , volume=. 2023 , number=

    work page 2023

  21. [21]

    Equations in simple Lie algebras , journal=

    Bandman, Tatiana and Gordeev, Nikolai and Kunyavski. Equations in simple Lie algebras , journal=. 2012 , pages=

    work page 2012

  22. [22]

    , title=

    Carter, Roger W. , title=

    work page

  23. [23]

    Seligman, G. B. , title=

    work page

  24. [24]

    Pacific J

    Steinberg, Robert , title=. Pacific J. Math. , volume=. 1961 , pages=

    work page 1961

  25. [25]

    , title=

    Steinberg, R. , title=. Can. J. Math. , volume=

    work page

  26. [26]

    and Kantor, W

    Guralnick, R. and Kantor, W. , title=. J. Algebra , volume=

    work page

  27. [27]

    Bois, Jean-Marie , title=. Math. Z. , volume=. 2009 , number=

    work page 2009

  28. [28]

    , title=

    Kuranishi, M. , title=. Nagoya Math. J. , volume=

    work page

  29. [29]

    , title=

    Ionescu, T. , title=. Linear Algebra Appl. , volume=

    work page

  30. [30]

    Linear Algebra Appl

    Cantor, Omer and Jezernik, Urban and Zozaya, Andoni , title=. Linear Algebra Appl. , volume=. 2025 , paper=

    work page 2025

  31. [31]

    and Lubotzky, Alexander , title=

    Kantor, William M. and Lubotzky, Alexander , title=. Geom. Dedicata , volume=. 1990 , number=

    work page 1990