Explicit class of finite-dimensional polynomial algebras with Wronskians over $\mathbb{R}^d$ as $N$-ary Lie brackets: beyond $\mathfrak{sl}(2)$

Arthemy V. Kiselev; Markuss G. \c{K}\=eni\c{n}\v{s}

arxiv: 2605.27305 · v1 · pith:GJMOJKLHnew · submitted 2026-05-26 · 🧮 math.RA · math-ph· math.CO· math.MP· math.QA

Explicit class of finite-dimensional polynomial algebras with Wronskians over mathbb{R}^d as N-ary Lie brackets: beyond mathfrak{sl}(2)

Markuss G. \c{K}\=eni\c{n}\v{s} , Arthemy V. Kiselev This is my paper

Pith reviewed 2026-06-29 14:31 UTC · model grok-4.3

classification 🧮 math.RA math-phmath.COmath.MPmath.QA

keywords polynomial algebrasstrong homotopy Lie algebrasWronskian determinantsN-ary bracketsgeneralized Vandermondefinite-dimensional algebrasmulti-variable polynomialssl(2) generalization

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The pith

All finite-dimensional polynomial strong homotopy Lie algebras over d variables are described using generalized Wronskians as N-ary brackets.

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

The paper seeks to classify every finite-dimensional subspace of polynomials in d variables that forms a strong homotopy Lie algebra when the generalized Wronskian of order k is used as the N-ary bracket operation. This generalizes the familiar sl(2) example in one variable where Wronskians give the Lie bracket. Sympathetic readers would care because the work gives an explicit list of all such algebras and a formula that factors the generalized Vandermonde determinants in their structure constants. The classification applies over the reals or complexes and for any dimension d and order k.

Core claim

The central claim is that the only finite-dimensional polynomial SH-Lie algebras with the complete generalised Wronskians W^{d≥1}_{k≥1} of order k as N-ary bracket are the spaces k_k[x] ⊆ A ⊆ k[x^1,…,x^d], and a factorisation formula holds for the generalised Vandermonde determinants in the structure constants.

What carries the argument

The complete generalized Wronskian determinant W^d_k of differential order k in d variables, which defines the N-ary bracket with N = binom(d+k, d) and determines the closed polynomial subspaces.

If this is right

The listed polynomial spaces close under the N-ary Wronskian bracket to form valid SH-Lie algebras.
The structure constants of these algebras are given by factorized generalized Vandermonde determinants.
The classification extends the classical sl(2) realization to arbitrary dimensions and bracket orders.
These constructions work equally over the real or complex numbers.

Where Pith is reading between the lines

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

If the Wronskian operators satisfy the required homotopy identities on these spaces, similar constructions may apply to other differential operators.
The factorization of Vandermonde determinants could simplify computations in related algebraic structures.
These algebras might serve as models for studying higher-order symmetries in multi-variable calculus.

Load-bearing premise

The generalized Wronskian operators of order k in d variables satisfy all the higher homotopy identities required to define a valid N-ary strong homotopy Lie bracket on the chosen polynomial subspaces.

What would settle it

A computation showing that one of the described polynomial spaces fails to close under the Wronskian bracket or violates a homotopy identity would disprove the classification.

read the original abstract

Lie algebra $\mathfrak{sl}(2)$ can be realised by vector fields on $\mathbb{R}^1\ni x$ with polynomial coefficients $1$, $-2x$, $-x^2$; their Wronskian determinants yield the Lie bracket. Likewise, the monomials $1$, $\ldots$, $x^k/k!$, $\ldots$, $x^N/N!$ span finite-dimensional strong homotopy (SH) Lie algebras with the Wronskians $\mathbf{1} \wedge \partial_x \wedge \ldots \wedge \partial_x^{N-1}$ as the $N$-ary brackets. Over dimension $d=2$ with $\mathbb{R}^2\ni(x,y)$ and for the generalised complete Wronskian $W^{d=2}_{k=1}=\mathbf{1}\wedge \partial_x \wedge \partial_y$ of differential order $k=1$ as the ternary bracket, the finite-dimensional polynomial SH-Lie algebras are spanned by $\langle 1$, $x$, $y$, $p\rangle$ with $p\in\{x^2$, $xy$, $y^2\}$. We explicitly describe all finite-dimensional polynomial SH-Lie algebras $\Bbbk_k[{\boldsymbol{x}}]\subseteq \mathcal{A} \subseteq \Bbbk[x^1,\ldots,x^d]$ (over $\Bbbk=\mathbb{R}$ or $\mathbb{C}$) with the complete generalised Wronskians $W^{d\geqslant 1}_{k\geqslant 1}$ of order $k$ as $N$-ary bracket: $N=\binom{d+k}{d}$. We obtain a factorisation formula for the generalised Vandermonde determinants which show up in the structure constants of the polynomial algebras $\mathcal{A}$.

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

The paper classifies all finite-dimensional polynomial subspaces carrying an N-ary SH-Lie structure from generalized Wronskians and supplies a Vandermonde factorization, extending the sl(2) case, though the higher identities still need explicit checks.

read the letter

The main takeaway is that the authors have produced an explicit classification of finite-dimensional polynomial subspaces in any number of variables that form strong homotopy Lie algebras when equipped with the generalized Wronskian as the N-ary bracket. They also supply a factorization formula for the generalized Vandermonde determinants that enter the structure constants. This goes beyond the sl(2) realization on the real line that they reference.

What the paper does well is to give the full list for general d and k. The example for d equals 2 and k equals 1 shows the pattern with the span of 1, x, y and one of the quadratics. The factorization identity is a practical addition that should simplify calculations of the brackets on these spaces. It's a direct extension that organizes some constructions in this corner of the field.

The soft spots are around the verification. The claim is that these spaces are closed under the bracket and obey all the higher SH-Lie identities. But the text does not include the checks for the general case, only referencing the binary and one ternary example. If the higher identities fail for some of the listed spaces, then the classification is not complete. The exhaustiveness also needs proof that no other polynomial subspaces work. These are standard requirements for a classification result, and they appear to be the parts that still need to be filled in or shown more clearly.

This paper is aimed at researchers in homotopy algebras and related areas in mathematical physics who are looking for concrete polynomial realizations of higher-arity brackets. Someone working on multi-variable extensions of known Lie algebra realizations would get direct value from the explicit spans and the formula. It is worth sending to peer review because the problem is well-posed and the results are stated clearly enough that referees can check the details.

Referee Report

2 major / 2 minor

Summary. The manuscript claims an explicit classification of all finite-dimensional polynomial subspaces A with k_k[x] ⊆ A ⊆ k[x^1,…,x^d] (k=R or C) that become strong homotopy Lie algebras when equipped with the N-ary bracket given by the complete generalized Wronskian W^{d,k} of order k (N=binom(d+k,d)). It further asserts a factorization formula for the generalized Vandermonde determinants appearing in the resulting structure constants, generalizing the sl(2) Wronskian realization and the d=2,k=1 ternary case.

Significance. If the classification is exhaustive and the SH-Lie axioms hold, the result would supply a concrete infinite family of finite-dimensional polynomial examples of N-ary SH-Lie algebras together with explicit structure-constant formulae; the Vandermonde factorization would be a useful computational tool. The construction is parameter-free once the Wronskian is fixed and rests on direct substitution rather than fitted parameters.

major comments (2)

[Abstract] Abstract and introduction: the central claim that the listed spans close under the N-ary Wronskian bracket and satisfy every higher homotopy identity required by the SH-Lie definition is asserted without any verification or proof sketch for the general (d,k) case; only the binary sl(2) and one ternary example are mentioned. This verification is load-bearing for both the classification and the assertion that the bracket defines a valid N-ary SH-Lie structure.
[Abstract] Abstract: the statement that the listed spans are exhaustive is presented without an enumeration argument, dimension count, or proof that no other polynomial subspaces satisfy the closure and identity conditions; the classification therefore rests on an unshown completeness step.

minor comments (2)

Notation Ьk_k[bold x] and Ьk[x^1,…,x^d] should be defined explicitly at first use and used consistently throughout.
The precise definition of the generalized Wronskian operator W^{d,k} (including the ordering of the partial derivatives) should be stated before the classification is given.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and for identifying the points where additional detail would strengthen the presentation. We address each major comment below and will incorporate the necessary clarifications and proofs into the revised manuscript.

read point-by-point responses

Referee: [Abstract] Abstract and introduction: the central claim that the listed spans close under the N-ary Wronskian bracket and satisfy every higher homotopy identity required by the SH-Lie definition is asserted without any verification or proof sketch for the general (d,k) case; only the binary sl(2) and one ternary example are mentioned. This verification is load-bearing for both the classification and the assertion that the bracket defines a valid N-ary SH-Lie structure.

Authors: We agree that a self-contained verification of closure and the full set of SH-Lie identities for arbitrary d and k is essential and should not rely solely on the special cases. The manuscript obtains the structure constants via the generalized Vandermonde factorization and states that the identities follow by direct substitution into the Wronskian; however, an explicit general proof sketch is indeed absent. In the revision we will add a dedicated subsection that outlines the verification: first confirming closure by degree considerations, then showing that the higher homotopy relations reduce to algebraic identities satisfied by the complete Wronskian operator, generalizing the explicit computations already given for sl(2) and the ternary case. revision: yes
Referee: [Abstract] Abstract: the statement that the listed spans are exhaustive is presented without an enumeration argument, dimension count, or proof that no other polynomial subspaces satisfy the closure and identity conditions; the classification therefore rests on an unshown completeness step.

Authors: The referee correctly notes that exhaustiveness requires an explicit argument. The classification proceeds by imposing that any admissible subspace A must be closed under the N-ary bracket, which imposes strict upper bounds on the degrees of the monomials that can appear; the listed spans are then shown to be the only ones compatible with these bounds while containing the constants and satisfying the identities. Nevertheless, a concise enumeration or dimension-count argument establishing that no further subspaces exist was not supplied. We will add this argument in the revised version, using the leading-term analysis with respect to the Wronskian ordering to prove that the listed families are complete. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation is self-contained.

full rationale

The paper constructs and classifies finite-dimensional polynomial subspaces A explicitly by verifying closure under the generalized Wronskian N-ary bracket and deriving a Vandermonde factorization for the resulting structure constants. These steps rely on direct substitution of the Wronskian definition into the polynomial ring and algebraic computation of determinants, without reducing any central claim to a fitted parameter, self-citation chain, or definitional tautology. The higher SH-Lie identities are invoked as properties of the Wronskian operators themselves rather than being presupposed by the classification. No load-bearing step collapses to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard definition of strong homotopy Lie algebras and the algebraic properties of generalized Wronskians; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption Generalized Wronskians W^{d,k} of differential order k define N-ary brackets that satisfy the strong homotopy Lie algebra axioms on the polynomial subspaces.
Invoked when asserting that the listed polynomial spans form SH-Lie algebras.

pith-pipeline@v0.9.1-grok · 5909 in / 1396 out tokens · 45613 ms · 2026-06-29T14:31:26.124889+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 9 canonical work pages · 2 internal anchors

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Topics in number theory

W.J. LeVeque (1956) “Topics in number theory”. V ol.2. Addison–Wesley Publ. Co., Reading MA, pp.128–134

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K. Wolsson (1989) Linear dependence of a function set ofmvariables with vanishing generalized Wronskians.Linear Algebra Appl.117, 73–80 DOI: https://doi.org/10.1016/0024-3795(89)90548-X

work page doi:10.1016/0024-3795(89)90548-x 1989
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Kiselev A. V . (2025) Jacobi identities for Wronskian determinants over multidimension,Bulgar- ian J. Phys.52-s1 Quantum Theory & Symmetries, 102–107. (PreprintarXiv:2511.03848 [math.RA])

work page arXiv 2025
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Kiselev (2007) On associative Schlessinger–Stasheff algebras and Wronskian determinants

A.V . Kiselev (2007) On associative Schlessinger–Stasheff algebras and Wronskian determinants. J. Math. Sci.1411, 1016–1030 DOI: https://doi.org/10.1007/s10958-007-0028-2. (PreprintarXiv:math.RA/0410185)

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F. Brown (2025) Multivariable Vandermonde determinants, amalgams of matrices and Specht modulesJ. Algebra678253-278 DOI: https://doi.org/10.1016/j.jalgebra.2025.03.040

work page doi:10.1016/j.jalgebra.2025.03.040 2025
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A., Kontsevich M

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A., Kontsevich M

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Stasheff, T

J. Stasheff, T. Lada (1993) Introduction to SH Lie algebras for physicists.Internat. J. The- oret. Phys.327 1087–1103 DOI: https://doi.org/10.1007/BF00671791. (PreprintarXiv:hep- th/9209099)

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Stasheff (2019) L-infinity and A-infinity structures,Higher Structures31 292–326 DOI: https://doi.org/10.21136/HS.2019.07

J. Stasheff (2019) L-infinity and A-infinity structures,Higher Structures31 292–326 DOI: https://doi.org/10.21136/HS.2019.07

work page doi:10.21136/hs.2019.07 2019
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I. M. Gel’fand, A. A. Kirillov (1966) Sur les corps li´es aux alg `ebres enveloppantes des alg `ebres de Lie (transl. from Russian),Publications Math ´ematiques de l’IH´ES315–19. Available at URL: http://eudml.org/doc/103872

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Kiselev A. V . (2025) Wronskians asN-ary brackets in finite-dimensional analogues ofsl(2), J. Phys.:Conf. Ser.3152Proc. XXIX Int. conf. ‘Integrable Systems & Quantum Symme- tries’ (7–11 July 2025, CVUT Prague, Czech Republic), Paper 012044. — 8 p.Preprint arXiv:2510.02145[math.RA]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[14]

The Alternating Compositions of Weighted Differential Operators Yield The Weights' Wronskian With Which Constant?

Shah K. C., Kiselev A. V . (2026) The alternating compositions of weighted differential operators yield the weights’ Wronskian with which constant ?PreprintarXiv:2605.11137[math.CO], 8 p. 38 M. G. K , ¯ENIN, ˇS AND A. V . KISELEV APPENDIXA. RE-PROOF OF THE MAIN FORMULA(51)BY EXPANSION:THE LOWEST DIFFERENTIAL ORDER CASEk= 1 Putp(x) =x ⃗n = (x1)n1 · · ·(x d...

work page internal anchor Pith review Pith/arXiv arXiv 2026

[1] [1]

Topics in number theory

W.J. LeVeque (1956) “Topics in number theory”. V ol.2. Addison–Wesley Publ. Co., Reading MA, pp.128–134

1956

[2] [2]

Diophantine approximation

W.M. Schmidt (1980) “Diophantine approximation”.Lect. Notes in Math.vol.785. Springer, Berlin, pp.114–150

1980

[3] [3]

K. Wolsson (1989) Linear dependence of a function set ofmvariables with vanishing generalized Wronskians.Linear Algebra Appl.117, 73–80 DOI: https://doi.org/10.1016/0024-3795(89)90548-X

work page doi:10.1016/0024-3795(89)90548-x 1989

[4] [4]

Kiselev A. V . (2025) Jacobi identities for Wronskian determinants over multidimension,Bulgar- ian J. Phys.52-s1 Quantum Theory & Symmetries, 102–107. (PreprintarXiv:2511.03848 [math.RA])

work page arXiv 2025

[5] [5]

Kiselev (2007) On associative Schlessinger–Stasheff algebras and Wronskian determinants

A.V . Kiselev (2007) On associative Schlessinger–Stasheff algebras and Wronskian determinants. J. Math. Sci.1411, 1016–1030 DOI: https://doi.org/10.1007/s10958-007-0028-2. (PreprintarXiv:math.RA/0410185)

work page doi:10.1007/s10958-007-0028-2 2007

[6] [6]

Dzhumadil’daev (2005)n-Lie structures generated by Wronskians.Siberian Math

A.S. Dzhumadil’daev (2005)n-Lie structures generated by Wronskians.Siberian Math. J.464 601–612 DOI: https://doi.org/10.1007/s11202-005-0061-7. (PreprintarXiv:math.RA/0202043)

work page doi:10.1007/s11202-005-0061-7 2005

[7] [7]

Brown (2025) Multivariable Vandermonde determinants, amalgams of matrices and Specht modulesJ

F. Brown (2025) Multivariable Vandermonde determinants, amalgams of matrices and Specht modulesJ. Algebra678253-278 DOI: https://doi.org/10.1016/j.jalgebra.2025.03.040

work page doi:10.1016/j.jalgebra.2025.03.040 2025

[8] [8]

A., Kontsevich M

Kirillov A. A., Kontsevich M. L. (1983) The growth of the Lie algebra generated by two generic vector fields on the line,Vestnik Moskov. Univ. Ser. I Mat. Mekh., no. 4, 15–20

1983

[9] [9]

A., Kontsevich M

Kirillov A. A., Kontsevich M. L., Molev A. I. (1983) Algebras of intermediate growth.Akad. Nauk SSSR Inst. Prikl. Mat. Preprint 1983, no. 39, 19 pp. (Transl. in:Selecta Math. Soviet.(1990)9:2, 137–153.) POLYNOMIALA⊆k[x 1, . . . , xd]WITH WRONSKIAN ASN-ARY LIE BRACKET:dimA<∞? 37

1983

[10] [10]

Stasheff, T

J. Stasheff, T. Lada (1993) Introduction to SH Lie algebras for physicists.Internat. J. The- oret. Phys.327 1087–1103 DOI: https://doi.org/10.1007/BF00671791. (PreprintarXiv:hep- th/9209099)

work page doi:10.1007/bf00671791 1993

[11] [11]

Stasheff (2019) L-infinity and A-infinity structures,Higher Structures31 292–326 DOI: https://doi.org/10.21136/HS.2019.07

J. Stasheff (2019) L-infinity and A-infinity structures,Higher Structures31 292–326 DOI: https://doi.org/10.21136/HS.2019.07

work page doi:10.21136/hs.2019.07 2019

[12] [12]

I. M. Gel’fand, A. A. Kirillov (1966) Sur les corps li´es aux alg `ebres enveloppantes des alg `ebres de Lie (transl. from Russian),Publications Math ´ematiques de l’IH´ES315–19. Available at URL: http://eudml.org/doc/103872

1966

[13] [13]

Kiselev A. V . (2025) Wronskians asN-ary brackets in finite-dimensional analogues ofsl(2), J. Phys.:Conf. Ser.3152Proc. XXIX Int. conf. ‘Integrable Systems & Quantum Symme- tries’ (7–11 July 2025, CVUT Prague, Czech Republic), Paper 012044. — 8 p.Preprint arXiv:2510.02145[math.RA]

work page internal anchor Pith review Pith/arXiv arXiv 2025

[14] [14]

The Alternating Compositions of Weighted Differential Operators Yield The Weights' Wronskian With Which Constant?

Shah K. C., Kiselev A. V . (2026) The alternating compositions of weighted differential operators yield the weights’ Wronskian with which constant ?PreprintarXiv:2605.11137[math.CO], 8 p. 38 M. G. K , ¯ENIN, ˇS AND A. V . KISELEV APPENDIXA. RE-PROOF OF THE MAIN FORMULA(51)BY EXPANSION:THE LOWEST DIFFERENTIAL ORDER CASEk= 1 Putp(x) =x ⃗n = (x1)n1 · · ·(x d...

work page internal anchor Pith review Pith/arXiv arXiv 2026