Explicit class of finite-dimensional polynomial algebras with Wronskians over $\mathbb{R}^d$ as $N$-ary Lie brackets: beyond $\mathfrak{sl}(2)$
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866918524802301952 |
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| author | Ķēniņš, Markuss G. Kiselev, Arthemy V. |
| author_facet | Ķēniņš, Markuss G. Kiselev, Arthemy V. |
| contents | 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 $\mathcal{A}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27305 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Explicit class of finite-dimensional polynomial algebras with Wronskians over $\mathbb{R}^d$ as $N$-ary Lie brackets: beyond $\mathfrak{sl}(2)$ Ķēniņš, Markuss G. Kiselev, Arthemy V. Rings and Algebras Mathematical Physics Combinatorics Quantum Algebra 05E16, 11B39, 15A15, 17A42 (primary) 05A10, 05A19, 17B65, 58H15 (secondary) 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 $\mathcal{A}$. |
| title | Explicit class of finite-dimensional polynomial algebras with Wronskians over $\mathbb{R}^d$ as $N$-ary Lie brackets: beyond $\mathfrak{sl}(2)$ |
| topic | Rings and Algebras Mathematical Physics Combinatorics Quantum Algebra 05E16, 11B39, 15A15, 17A42 (primary) 05A10, 05A19, 17B65, 58H15 (secondary) |
| url | https://arxiv.org/abs/2605.27305 |