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

Based on the talks given by the first author at the Algebra seminar (Bernoulli Institute, Groningen, the Netherlands) and by the last author at the Prague Mathematical Physics seminar (Charles University, Czech Republic) and at the Mathematics seminar (IHÉS, Bures-sur-Yvette, France); 44 pages, 1 figure, 4 appendices