Jacobi identities for Wronskian determinants over multidimension

Abstract

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

Based on the talk given at the XIII International symposium on Quantum Theory and Symmetries -- QTS13 (Yerevan, Armenia, 28 July -- 1 August 2025); 6 pages