Exceptionally simple PDE

TL;DR

This paper provides explicit PDE models with large symmetry algebras related to parabolic contact structures, generalizing classical models and offering formulas for harmonic curvature in complex Lie algebra contexts.

Contribution

It introduces a uniform approach to describing PDEs with symmetries linked to all complex simple Lie algebras except sl(2), extending classical models and analyzing curvature.

Findings

Explicit PDE models with symmetry algebras isomorphic to complex simple Lie algebras

Formulas for harmonic curvature of G2-contact structures

Descriptions of submaximally symmetric models for G-contact structures

Abstract

We give local descriptions of parabolic contact structures and show how their flat models yield explicit PDE having symmetry algebras isomorphic to all complex simple Lie algebras except $\mathfrak{sl}_2$. This yields a remarkably uniform generalization of the Cartan-Engel models from 1893 in the $G_2$ case. We give a formula for the harmonic curvature of a $G_2$-contact structure and describe submaximally symmetric models for general $G$-contact structures.