Jacobi identities for Wronskian determinants over multidimension

Abstract

The generalised Wronskian of differential order $k\geqslant 1$ for $N$ functions $f_1$, $\ldots$, $f_N$ in $d\geqslant 1$ independent variables $x^1$, $\ldots$, $x^d$ is the determinant of the matrix with these functions' derivatives $\partial^{|\sigma_i|} f_j / \partial (x^1)^{\sigma_i^1}\cdots \partial (x^d)^{\sigma_i^d}$ (of orders $0 \leqslant |\sigma_i| \leqslant k$), where the multi-indices $\sigma_i$ mark (all or part of) fibre variables $u_{\sigma_i}$ in the $k$th jet space $J^k\bigl(\mathbb{R}^d\to\mathbb{R}\bigr)$. We prove that these (in)complete Wronskians -- provided that their lowest-order parts are complete at differential orders $\ell\leqslant 1$ -- over the $d$-dimensional base satisfy the table of bi-linear, Jacobi-type identities for Schlessinger--Stasheff's strongly homotopy Lie algebras.