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

Lie algebra $\mathfrak{s}\mathfrak{l}(2)$ 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 $1$, $\dots$, $x^k\mathrm{/}k!$, $\dots$, $x^N\mathrm{/}N!$ span finite-dimensional strong homotopy (SH) Lie algebras with the Wronskians $\mathbf{1}\wedge {\mathrm{\partial }}_x\wedge \dots \wedge {\mathrm{\partial }}_x^{N-1}$ as the $N$-ary brackets. Over dimension $d=2$ with ${\mathbb{R}}^2\ni (x,y)$ and for the complete generalised Wronskian $W_{d=2}^{k=1}=\mathbf{1}\wedge {\mathrm{\partial }}_x\wedge {\mathrm{\partial }}_y$ of differential order $k=1$ as the ternary bracket, the finite-dimensional polynomial SH-Lie algebras are spanned by $⟨1$, $x$, $y$, $p⟩$ with $p\in \{x^2$, $xy$, $y^2\}$. We explicitly describe all finite-dimensional polynomial SH-Lie algebras ${\mathbb{k}}_k[\mathit{x}]\subseteq \mathcal{A}\subseteq \mathbb{k}[x^1,\dots ,x^d]$ (over $\mathbb{k}=\mathbb{R}$ or $\mathbb{C}$) with the complete generalised Wronskians $W_{d⩾1}^{k⩾1}$ of order $k$ as $N$-ary brackets: $N=\left(\genfrac{}{}{0px}{}{d+k}{d}\right)$. We obtain a factorisation formula for the generalised Vandermonde determinants which show up in the structure constants of the polynomial algebras $\mathcal{A}$.