Exceptionally simple PDE

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 <math><msub is="true"><mrow is="true"><mi mathvariant="fraktur" is="true">sl</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub></math>. This yields a remarkably uniform generalization of the Cartan–Engel models from 1893 in the <math><msub is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub></math> case. We give a formula for the harmonic curvature of a <math><msub is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub></math>-contact structure and describe submaximally symmetric models for general G-contact structures.