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

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 $\mathcal{A}$.