bibkey: keninskiselev2026wronskianalgebras
authors: Markuss G. Ķēniņš; Arthemy V. Kiselev
year: 2026
title: "Explicit class of finite-dimensional polynomial algebras with Wronskians over R^d as N-ary Lie brackets: beyond sl(2)"
doi: null
url: https://arxiv.org/abs/2605.27305v2
claim: "The paper studies finite-dimensional polynomial N-ary SH-Lie algebras in d variables with the complete generalised Wronskian of order k as bracket, shows that with a monomial of degree k+1 on top of the polynomials of degree at most k the algebra is perfect only for d=1 or d=2, k=1, gives perfect examples with homogeneous polynomials for d=2,3 and k=1, and asks whether for every d>=2 and k>=1 some homogeneous polynomial of degree k+1 makes the algebra perfect."
strata_touched:
  - D5/S3/HomologicalAlgebra/WronskianPerfectLonelyAlgebra
license: citation-only
triage: anchor

Explicit class of finite-dimensional polynomial algebras with Wronskians

M. G. Ķēniņš and A. V. Kiselev, arXiv:2605.27305v2 (10 July 2026; v1 26 May 2026). Quotations are from the v2 TeX source, with the macros \Bbbk, \bx, \EuA, \Bone and \wedges written as $\mathbb{k}$, $\mathbf{x}$, $\mathcal{A}$, $1$ and $\wedge \cdots \wedge$, and cross-references omitted.

Notation: $\mathbb{k}=\mathbb{R}$ or $\mathbb{C}$ is the base field; ${\mathbb{k}}_k[x^1,\dots ,x^d]$ denotes the polynomials of degree at most $k$; for a multi-index $\vec{r}\in {\mathbb{N}}_{\ge 0}^d$, ${\partial }^{|\vec{r}|}/\partial {\mathbf{x}}^{\vec{r}}={\partial }_{x^1}^{r^1}\cdots {\partial }_{x^d}^{r^d}$.

The bracket:

The $N$-linear (over $\mathbb{k}$) totally antisymmetric, $C^{\infty }({\mathbb{R}}^d)$-valued differential operator $W_d^k=1\wedge {\partial }_{\mathbf{x}}\wedge \cdots \wedge {\partial }_{\mathbf{x}\dots \mathbf{x}}^k$ is the complete generalised Wronskian determinant of differential order $k$ over dimension $d$.

Here $N=\left(\genfrac{}{}{0ex}{}{d+k}{k}\right)$, and the rows of the determinant are the operators ${\partial }^{|\vec{r}|}/\partial {\mathbf{x}}^{\vec{r}}$ with $|\vec{r}|\le k$, each once.

Perfectness:

we say that the polynomial $N$-ary SH-Lie algebra $\mathcal{A}$ with the Wronskian $W_{d⩾1}^{k⩾1}$ as the bracket is perfect if the bracket spans the whole algebra: ${\mathrm{Span}}_{\mathbb{k}}{W}_d^k(\mathcal{A},\dots ,\mathcal{A})=\mathcal{A}$.

Examples: $⟨1,x,y,xy⟩$ for $d=2$, $k=1$, and $⟨1,x,y,z,xy+yz+zx⟩$ for $d=3$, $k=1$, are perfect; with a monomial of degree $k+1$ on top of ${\mathbb{k}}_k[\mathbf{x}]$ the algebra is perfect only for $d=1$ or for $d=2$, $k=1$.

The open problem:

Let the base dimension $d⩾2$ and differential order $k⩾1$ be arbitrary. Put the complete generalised Wronskian $W_d^k$ as the $N$-ary bracket, $N=\left(\genfrac{}{}{0ex}{}{d+k}{d}\right)$, on the $N$-ary SH-Lie polynomial algebra $\mathbb{k}[x^1,\dots ,x^d]$. Does there exist an homogeneous polynomial $p(\mathbf{x})\in \mathbb{k}[\mathbf{x}]$ of degree $k+1$ such that the $N$-ary SH-Lie polynomial subalgebra $\mathcal{A}={\mathbb{k}}_k[\mathbf{x}]\oplus ⟨p(\mathbf{x})⟩$ is perfect w.r.t the Wronskian, i.e. ${\mathrm{Span}}_{\mathbb{k}}{W}_d^k(\mathcal{A},\dots ,\mathcal{A})=\mathcal{A}$?

Verified locator

Source: https://arxiv.org/abs/2605.27305v2

• URL: https://arxiv.org/abs/2605.27305v2 (the e-print is a tar archive; member FinDim220526v2.tex, 2221 lines, md5 777135f176c9db701f21e5554216fb33):
  • perfectness, Definition def:perfect-lie-algebra, l. 517–521;
  • notation, l. 538–542 and Notation not:multi-index, l. 613–623;
  • the Wronskian, Definition def:wronskian, l. 625–627;
  • the example for $d=2$, Example ex:dimension-two-closed-old, l. 467–478;
  • monomial imperfection, Proposition prop:imperfection, l. 1198;
  • the example for $d=3$, Example ex:perfection, l. 1247–1255;
  • the open problem, Open problem open:perfect-algebras, l. 1258–1260.
• The author list and dates are from the TeX \author fields and the arXiv metadata; Crossref lists no journal version.