Jacobi identities for Wronskian determinants over multidimension

TL;DR

This paper proves that generalized Wronskian determinants of multiple functions over multiple variables satisfy Jacobi identities related to homotopy Lie algebras, extending classical identities to higher dimensions and differential orders.

Contribution

It establishes Jacobi identities for multidimensional Wronskians, connecting them to strongly homotopy Lie algebra structures, a novel extension of classical Wronskian properties.

Findings

Generalized Wronskians satisfy bi-linear Jacobi identities.

Identities hold under conditions on lowest-order parts of Wronskians.

Extends classical Wronskian theory to multidimensional differential settings.

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.