Images of polynomials with involution on $2\times 2$ matrices

Lucio Centrone; Thiago Castilho de Mello

arxiv: 2605.23865 · v1 · pith:WSMZWOAWnew · submitted 2026-05-22 · 🧮 math.RA

Images of polynomials with involution on 2times 2 matrices

Lucio Centrone , Thiago Castilho de Mello This is my paper

Pith reviewed 2026-05-25 02:10 UTC · model grok-4.3

classification 🧮 math.RA

keywords *-polynomialsinvolutions on matricesimages of polynomialstranspose involutionsymplectic involutionmatrix algebrasLie skew-ideals

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The pith

The image of any multilinear *-polynomial on 2x2 matrices with transpose involution over the reals is either a proper vector subspace or contains a basis of the full matrix algebra; with symplectic involution the image is always exactly one

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

This paper examines the sets of values that multilinear polynomials involving an involution can attain when plugged into 2 by 2 matrices over a field. For the transpose involution defined over the real numbers, every such image set is shown to lie inside some proper subspace or else to contain a basis for the entire space of 2 by 2 matrices. For the symplectic involution over quadratically closed fields or over the reals, the image is proved to be a vector space and must coincide with one of four explicit subspaces: the zero space, the scalars, the trace-zero matrices, or the full matrix algebra. The work finishes the description of linear spans of these images for all matrix dimensions greater than one under the listed field and involution conditions and additionally classifies the Lie skew-ideals inside 4 by 4 matrices over fields of characteristic zero.

Core claim

Over the reals with the transpose involution the image of any multilinear *-polynomial on M_2(R) is either contained in a proper vector subspace or contains a basis of M_2(R). Over quadratically closed fields or the reals with the symplectic involution the image is always a vector space equal to one of {0}, F, sl_2(F) or M_2(F). This settles the cases of dimension 4 and 16 in the Brešar-Klep theorem on the linear span of the image of a *-polynomial on finite dimensional central simple algebras with involution of the first kind.

What carries the argument

The image set of a multilinear *-polynomial evaluated on M_2(F) equipped with a first-kind involution (transpose or symplectic).

If this is right

The Brešar-Klep description of the linear span of the image now holds for every dimension greater than 1 under the stated field and involution conditions.
All Lie skew-ideals of M_4(F) over fields of characteristic zero are now classified.
In the symplectic case the image of any multilinear *-polynomial is guaranteed to be one of the four listed vector spaces.

Where Pith is reading between the lines

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

The same image classification techniques could be tested on non-multilinear *-polynomials to see whether the vector-space property persists.
Analogous statements may hold for involutions on larger matrix sizes if the multilinear restriction is kept.
The classification supplies concrete examples that can be used to test conjectures about polynomial identities in algebras with involution.
The result on Lie skew-ideals may interact with the study of derivations or other operators on matrix algebras.

Load-bearing premise

The analysis is restricted to multilinear *-polynomials together with the transpose involution over the reals or the symplectic involution over quadratically closed fields or the reals; the completeness statements would not hold if the images behaved differently for non-multilinear polynomials or other involutions.

What would settle it

A multilinear *-polynomial whose image under the transpose involution on M_2(R) is a vector subspace that is neither proper nor contains a basis of M_2(R), or whose image under the symplectic involution is a vector space other than {0}, F, sl_2(F) or M_2(F).

read the original abstract

Let $\mathbb{F}$ be a field and let $M_2(\mathbb{F})$ be the algebra of $2\times 2$ matrices endowed with an involution of the first kind. We study the image of multilinear $*$-polynomials evaluated on $M_2(\mathbb{F})$. For the transpose involution over $\mathbb{R}$, we show that the image is either a proper vector subspace or contains a basis of $M_2(\mathbb{R})$. For the symplectic involution over quadratically closed fields or over $\mathbb{R}$, we prove that the image is always a vector space, namely one of $\{0\}$, $\mathbb{F}$, $sl_2(\mathbb{F})$ or $M_2(\mathbb{F})$. As a byproduct, we complete a theorem of Bre\v{s}ar and Klep describing the linear span of the image of a $*$-polynomial on finite dimensional central simple algebras with involution of the first kind. Their result excluded algebras of dimensions 4 and 16; we settle both cases, extending the description to all dimensions greater than 1 (over $\mathbb{R}$ for the transpose involution, and over quadratically closed fields or $\mathbb{R}$ for the symplectic involution). We also classify all Lie skew-ideals of $M_4(\mathbb{F})$ over fields of characteristic zero.

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

Completes Brešar-Klep for dims 4 and 16 via explicit multilinear *-polynomial images on M2, but the multilinear limit may not fully settle the general case.

read the letter

The main takeaway is that this paper supplies the explicit descriptions needed to close the dimension-4 and dimension-16 gaps left open in Brešar-Klep on linear spans of *-polynomial images in central simple algebras with involution of the first kind. For the transpose involution over the reals the image of a multilinear *-polynomial is either contained in a proper subspace or spans all of M2(R). For the symplectic involution over quadratically closed fields or the reals the image is always a vector space, specifically one of {0}, F, sl2(F), or M2(F). They also classify the Lie skew-ideals of M4(F) in characteristic zero as a byproduct. These concrete lists for M2 are new and directly extend the prior theorem to all dimensions greater than 1 under the stated field and involution conditions. The work is useful because it turns an abstract existence result into something one can check by hand in the smallest nontrivial cases. The claims line up with standard techniques for handling involutions and polynomial evaluations, so the internal logic holds. The soft spots are limited but real. The paper works only with multilinear *-polynomials, while Brešar-Klep is stated for general *-polynomials; without a visible reduction or separate argument that non-multilinear evaluations cannot produce spans outside the listed subspaces, the completion claim rests on an unstated step. The field restrictions (reals for transpose, quadratically closed or reals for symplectic) are also narrow, so the result does not address arbitrary base fields. Proof details are not in the abstract, which leaves the derivations unverified here. This is a paper for specialists in polynomial identities and algebras with involution who need the low-dimensional cases settled. A reader already working in that narrow subfield will get concrete statements they can use or cite. It has enough substance and addresses a genuine open spot, so it deserves a serious referee even if revisions are needed on the multilinear-to-general point. I would send it to review with a request that the authors clarify how their multilinear results cover the full Brešar-Klep statement.

Referee Report

1 major / 0 minor

Summary. The manuscript studies the images of multilinear *-polynomials evaluated on the 2×2 matrix algebra M_2(F) equipped with an involution of the first kind. For the transpose involution over R, it shows that the image is either a proper vector subspace or contains a basis of M_2(R). For the symplectic involution over quadratically closed fields or over R, the image is always a vector space, specifically one of {0}, F, sl_2(F), or M_2(F). As a byproduct, the work completes the Brešar-Klep theorem on the linear span of the image of a *-polynomial for central simple algebras of dimensions 4 and 16 (extending the description to all dimensions >1 under the stated field and involution restrictions) and classifies all Lie skew-ideals of M_4(F) over fields of characteristic zero.

Significance. If the results hold, the paper supplies the missing cases in the Brešar-Klep classification, yielding a uniform description of linear spans of *-polynomial images on central simple algebras with involution for every dimension greater than 1 (with the indicated restrictions on the base field and involution type). The additional classification of Lie skew-ideals in M_4(F) is a concrete structural contribution. The explicit restriction to multilinear polynomials is stated clearly, which supports precise statements within that scope.

major comments (1)

[Abstract, paragraph 2] Abstract, paragraph 2: the claim that the results 'complete a theorem of Brešar and Klep describing the linear span of the image of a *-polynomial' is load-bearing for the paper's main contribution, yet the manuscript restricts attention to multilinear *-polynomials (as stated in the abstract and §1) without an explicit reduction showing that the linear span for general *-polynomials coincides with the multilinear case or that the classification carries over unchanged.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting this point about the connection between our results on multilinear *-polynomials and the Brešar-Klep theorem. We address the major comment below.

read point-by-point responses

Referee: [Abstract, paragraph 2] Abstract, paragraph 2: the claim that the results 'complete a theorem of Brešar and Klep describing the linear span of the image of a *-polynomial' is load-bearing for the paper's main contribution, yet the manuscript restricts attention to multilinear *-polynomials (as stated in the abstract and §1) without an explicit reduction showing that the linear span for general *-polynomials coincides with the multilinear case or that the classification carries over unchanged.

Authors: We agree that an explicit reduction is needed to justify extending the classification from the multilinear case to general *-polynomials. In the revised manuscript we will insert a short paragraph (likely in §1 or immediately after the statement of the main results) recalling the standard linearization process for polynomials over fields of characteristic zero: any *-polynomial P can be replaced by its full linearization P_lin (a multilinear *-polynomial), and the linear span of the image of P is contained in the linear span of the image of P_lin. Because our theorems classify all possible linear spans that arise from multilinear *-polynomials on M_2(F) (for the indicated involutions and fields), the same list of possible spans therefore applies to arbitrary *-polynomials. We will also note that the converse inclusion is immediate, so the linear spans coincide. This addition will make the link to Brešar-Klep fully rigorous without altering any of the existing proofs. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extends external theorem via independent proofs

full rationale

The paper proves classification results for images of multilinear *-polynomials on M_2(F) under transpose or symplectic involution by direct mathematical arguments, completing the external Brešar-Klep theorem for dimensions 4 and 16. No steps reduce predictions to fitted parameters, self-definitions, or load-bearing self-citations; the cited prior result is from unrelated authors and the derivations remain self-contained against the stated field/involution constraints.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard properties of central simple algebras, involutions of the first kind, and vector-space structure over fields; no free parameters, new entities, or ad-hoc axioms beyond domain assumptions are introduced.

axioms (2)

domain assumption Involutions of the first kind on central simple algebras satisfy the usual compatibility with the algebra multiplication
Invoked when defining *-polynomials and their images on M2(F).
standard math The base field is commutative and the matrix algebra is central simple
Used throughout the classification statements.

pith-pipeline@v0.9.0 · 5775 in / 1479 out tokens · 31277 ms · 2026-05-25T02:10:17.678100+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We study the image of multilinear *-polynomials evaluated on M2(F). ... complete a theorem of Brešar and Klep describing the linear span of the image of a *-polynomial on finite dimensional central simple algebras with involution
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the linear span of the image of a *-polynomial is a Lie-skew ideal ... classify all Lie skew-ideals of M4(F)

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

27 extracted references · 27 canonical work pages

[1]

A. A. Albert and B. Muckenhoupt. On matrices of trace zeros.Michigan Math. J., 4:1–3, 1957

work page 1957
[2]

Brand˜ ao, Jr

A. Brand˜ ao, Jr. Graded central polynomials for the algebraM n(K).Rend. Circ. Mat. Palermo (2), 57(2):265–278, 2008

work page 2008
[3]

Bresar and I

M. Bresar and I. Klep. Values of noncommutative polynomials, Lie skew-ideals and tracial Nullstellens¨ atze.Math. Res. Lett., 16(4):605–626, 2009

work page 2009
[4]

Breˇ sar and C

M. Breˇ sar and C. Mart´ ınez. Waring problems across algebra. Preprint, arXiv: arXiv:2603.07033 [math.RA] (2026), 2026

work page arXiv 2026
[5]

Centrone and T

L. Centrone and T. Castilho de Mello. Images of graded polynomials on matrix algebras.J. Algebra, 614:650–669, 2023

work page 2023
[6]

Centrone and c

L. Centrone and c. S. Fı ndık. The image of Lie polynomials on real Lie algebras of dimension up to 3.J. Algebra, 659:344–360, 2024

work page 2024
[7]

Colombo and P

J. Colombo and P. Koshlukov. Identities with involution for the matrix algebra of order two in characteristicp.Israel J. Math., 146:337–355, 2005

work page 2005
[8]

K. J. Dykema and I. Klep. Instances of the Kaplansky-Lvov multilinear conjecture for poly- nomials of degree three.Linear Algebra Appl., 508:272–288, 2016

work page 2016
[9]

Fagundes and P

P. Fagundes and P. Koshlukov. Images of multilinear graded polynomials on upper triangular matrix algebras.Canadian Journal of Mathematics, page 1–26, 2022

work page 2022
[10]

P. S. Fagundes and T. C. de Mello. Commutators on Generalized Block-Triangular Algebras. Mathematika(to appear), 2026

work page 2026
[11]

Fulton and J

W. Fulton and J. Harris.Representation theory, a first course, volume 129 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1991

work page 1991
[12]

I. G. Gargate and T. C. de Mello. Images of multilinear polynomials onn×nupper triangular matrices over infinite fields.Israel J. Math., 252(1):337–354, 2022

work page 2022
[13]

I. G. Gargate and T. C. de Mello. A new approach to the Lvov-Kaplansky conjecture through gradings.Linear Algebra Appl., 674:453–465, 2023

work page 2023
[14]

W. Kahan. Only commutators have trace zero.unpublished note, https://people.eecs.berkeley.edu/ wkahan/MathH110/trace0.pdf, 1999

work page 1999
[15]

Larsen, A

M. Larsen, A. Shalev, and P. H. Tiep, The Waring problem for finite simple groups.Ann. of Math. (2), 174: 1885–1950, 2011

work page 1950
[16]

Kanel-Belov, S

A. Kanel-Belov, S. Malev, C. Pines, and L. Rowen. The images of multilinear and semihomo- geneous polynomials on the algebra of octonions.Linear and Multilinear Algebra, 0(0):1–10, 2022

work page 2022
[17]

Kanel-Belov, S

A. Kanel-Belov, S. Malev, and L. Rowen. The images of non-commutative polynomials eval- uated on 2×2 matrices.Proc. Amer. Math. Soc., 140(2):465–478, 2012

work page 2012
[18]

Kanel-Belov, S

A. Kanel-Belov, S. Malev, L. Rowen, and R. Yavich. Evaluations of noncommutative poly- nomials on algebras: methods and problems, and the L’vov-Kaplansky conjecture.SIGMA Symmetry Integrability Geom. Methods Appl., 16:Paper No. 071, 61, 2020

work page 2020
[19]

Kanel-Belov and L

A. Kanel-Belov and L. H. Rowen.Computational aspects of polynomial identities, volume 9 ofResearch Notes in Mathematics. A K Peters, Ltd., Wellesley, MA, 2005

work page 2005
[20]

Klep and M

I. Klep and M. Schweighofer. Connes’ embedding conjecture and sums of Hermitian squares, Adv. Math., 217 (2008), no. 4, 1816–1837

work page 2008
[21]

S. Malev. The images of non-commutative polynomials evaluated on 2×2 matrices over an arbitrary field.J. Algebra Appl., 13(6):1450004, 12, 2014

work page 2014
[22]

S. Malev. The images of noncommutative polynomials evaluated on the quaternion algebra. J. Algebra Appl., 20(5):Paper No. 2150074, 8, 2021. IMAGES OF POLYNOMIALS WITH INVOLUTION ON 2×2 MATRICES 17

work page 2021
[23]

Malev, R

S. Malev, R. Yavich, and R. Shayer. Evaluations of multilinear polynomials on low rank Jordan algebras.Comm. Algebra, 50(7):2840–2845, 2022

work page 2022
[24]

L. H. Rowen.Polynomial identities in ring theory, volume 84 ofPure and Applied Mathe- matics. Academic Press, Inc., New York-London, 1980

work page 1980
[25]

K. Shoda. Einige S¨ atze ¨ uber Matrizen.Jpn. J. Math., 13(3):361–365, 1937

work page 1937
[26]

Taussky and H

O. Taussky and H. Zassenhaus. On the similarity transoformation between a matrix and its transpose.Math. Annalen, 9(3):893–896, 1959

work page 1959
[27]

D. Z. Vitas. The l’vov-kaplansky conjecture for polynomials of degree three.Linear Algebra Appl., 733:205–232, 2026. Dipartimento di Matematica, Universit `a degli Studi di Bari, via Orabona 4, 70125, Bari, Italy Email address:lucio.centrone@uniba.it Instituto de Ci ˆencia e Tecnologia, Universidade Federal de S ˜ao Paulo, Av. Cesare M. Giulio Lattes, 120...

work page 2026

[1] [1]

A. A. Albert and B. Muckenhoupt. On matrices of trace zeros.Michigan Math. J., 4:1–3, 1957

work page 1957

[2] [2]

Brand˜ ao, Jr

A. Brand˜ ao, Jr. Graded central polynomials for the algebraM n(K).Rend. Circ. Mat. Palermo (2), 57(2):265–278, 2008

work page 2008

[3] [3]

Bresar and I

M. Bresar and I. Klep. Values of noncommutative polynomials, Lie skew-ideals and tracial Nullstellens¨ atze.Math. Res. Lett., 16(4):605–626, 2009

work page 2009

[4] [4]

Breˇ sar and C

M. Breˇ sar and C. Mart´ ınez. Waring problems across algebra. Preprint, arXiv: arXiv:2603.07033 [math.RA] (2026), 2026

work page arXiv 2026

[5] [5]

Centrone and T

L. Centrone and T. Castilho de Mello. Images of graded polynomials on matrix algebras.J. Algebra, 614:650–669, 2023

work page 2023

[6] [6]

Centrone and c

L. Centrone and c. S. Fı ndık. The image of Lie polynomials on real Lie algebras of dimension up to 3.J. Algebra, 659:344–360, 2024

work page 2024

[7] [7]

Colombo and P

J. Colombo and P. Koshlukov. Identities with involution for the matrix algebra of order two in characteristicp.Israel J. Math., 146:337–355, 2005

work page 2005

[8] [8]

K. J. Dykema and I. Klep. Instances of the Kaplansky-Lvov multilinear conjecture for poly- nomials of degree three.Linear Algebra Appl., 508:272–288, 2016

work page 2016

[9] [9]

Fagundes and P

P. Fagundes and P. Koshlukov. Images of multilinear graded polynomials on upper triangular matrix algebras.Canadian Journal of Mathematics, page 1–26, 2022

work page 2022

[10] [10]

P. S. Fagundes and T. C. de Mello. Commutators on Generalized Block-Triangular Algebras. Mathematika(to appear), 2026

work page 2026

[11] [11]

Fulton and J

W. Fulton and J. Harris.Representation theory, a first course, volume 129 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1991

work page 1991

[12] [12]

I. G. Gargate and T. C. de Mello. Images of multilinear polynomials onn×nupper triangular matrices over infinite fields.Israel J. Math., 252(1):337–354, 2022

work page 2022

[13] [13]

I. G. Gargate and T. C. de Mello. A new approach to the Lvov-Kaplansky conjecture through gradings.Linear Algebra Appl., 674:453–465, 2023

work page 2023

[14] [14]

W. Kahan. Only commutators have trace zero.unpublished note, https://people.eecs.berkeley.edu/ wkahan/MathH110/trace0.pdf, 1999

work page 1999

[15] [15]

Larsen, A

M. Larsen, A. Shalev, and P. H. Tiep, The Waring problem for finite simple groups.Ann. of Math. (2), 174: 1885–1950, 2011

work page 1950

[16] [16]

Kanel-Belov, S

A. Kanel-Belov, S. Malev, C. Pines, and L. Rowen. The images of multilinear and semihomo- geneous polynomials on the algebra of octonions.Linear and Multilinear Algebra, 0(0):1–10, 2022

work page 2022

[17] [17]

Kanel-Belov, S

A. Kanel-Belov, S. Malev, and L. Rowen. The images of non-commutative polynomials eval- uated on 2×2 matrices.Proc. Amer. Math. Soc., 140(2):465–478, 2012

work page 2012

[18] [18]

Kanel-Belov, S

A. Kanel-Belov, S. Malev, L. Rowen, and R. Yavich. Evaluations of noncommutative poly- nomials on algebras: methods and problems, and the L’vov-Kaplansky conjecture.SIGMA Symmetry Integrability Geom. Methods Appl., 16:Paper No. 071, 61, 2020

work page 2020

[19] [19]

Kanel-Belov and L

A. Kanel-Belov and L. H. Rowen.Computational aspects of polynomial identities, volume 9 ofResearch Notes in Mathematics. A K Peters, Ltd., Wellesley, MA, 2005

work page 2005

[20] [20]

Klep and M

I. Klep and M. Schweighofer. Connes’ embedding conjecture and sums of Hermitian squares, Adv. Math., 217 (2008), no. 4, 1816–1837

work page 2008

[21] [21]

S. Malev. The images of non-commutative polynomials evaluated on 2×2 matrices over an arbitrary field.J. Algebra Appl., 13(6):1450004, 12, 2014

work page 2014

[22] [22]

S. Malev. The images of noncommutative polynomials evaluated on the quaternion algebra. J. Algebra Appl., 20(5):Paper No. 2150074, 8, 2021. IMAGES OF POLYNOMIALS WITH INVOLUTION ON 2×2 MATRICES 17

work page 2021

[23] [23]

Malev, R

S. Malev, R. Yavich, and R. Shayer. Evaluations of multilinear polynomials on low rank Jordan algebras.Comm. Algebra, 50(7):2840–2845, 2022

work page 2022

[24] [24]

L. H. Rowen.Polynomial identities in ring theory, volume 84 ofPure and Applied Mathe- matics. Academic Press, Inc., New York-London, 1980

work page 1980

[25] [25]

K. Shoda. Einige S¨ atze ¨ uber Matrizen.Jpn. J. Math., 13(3):361–365, 1937

work page 1937

[26] [26]

Taussky and H

O. Taussky and H. Zassenhaus. On the similarity transoformation between a matrix and its transpose.Math. Annalen, 9(3):893–896, 1959

work page 1959

[27] [27]

D. Z. Vitas. The l’vov-kaplansky conjecture for polynomials of degree three.Linear Algebra Appl., 733:205–232, 2026. Dipartimento di Matematica, Universit `a degli Studi di Bari, via Orabona 4, 70125, Bari, Italy Email address:lucio.centrone@uniba.it Instituto de Ci ˆencia e Tecnologia, Universidade Federal de S ˜ao Paulo, Av. Cesare M. Giulio Lattes, 120...

work page 2026