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 $1$, $-2x$, $-x<sup>2$; their Wronskian determinants yield the Lie bracket. Likewise, the monomials $1$, $\ldots$, $x<sup>k/k!$, $\ldots$, $x<sup>N/N!$ span finite-dimensional strong homotopy (SH) Lie algebras with the Wronskians $$\mathbf{1} \wedge \partial_x \wedge \ldots \wedge \partial_x<sup>{N-1}$$ as the $N$-ary brackets. Over dimension $d=2$ with $\mathbb{R}<sup>2\ni(x,y)$ 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 $p\in{x<sup>2$, $xy$, $y<sup>2}$. We explicitly describe all finite-dimensional polynomial SH-Lie algebras $$\Bbbk_k[{\boldsymbol{x}}]\subseteq \mathcal{A} \subseteq \Bbbk[x<sup>1,\ldots,x<sup>d]$$ (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 $\mathcal{A}$.