Lowest degree invariant 2nd order PDEs over rational homogeneous contact manifolds

Abstract

For each simple Lie algebra $\mathfrak{g}$ (excluding, for trivial reasons, type $\mathsf{C}$) we find the lowest possible degree of an invariant second-order PDE over the adjoint variety in $\mathbb{P}\mathfrak{g}$, a homogeneous contact manifold. Here a PDE $F(x^i,u,u_i,u_{ij})=0$ has degree $\le d$ if $F$ is a polynomial of degree $\le d$ in the minors of $(u_{ij})$, with coefficients functions of the contact coordinates $x^i$, $u$, $u_i$ (e.g., Monge-Ampère equations have degree 1). For $\mathfrak{g}$ of type $\mathsf{A}$ or $\mathsf{G}$ we show that this gives all invariant second-order PDEs. For $\mathfrak{g}$ of type $\mathsf{B}$ and $\mathsf{D}$ we provide an explicit formula for the lowest-degree invariant second-order PDEs. For $\mathfrak{g}$ of type $\mathsf{E}$ and $\mathsf{F}$ we prove uniqueness of the lowest-degree invariant second-order PDE; we also conjecture that uniqueness holds in type $\mathsf{D}$.

to appear on "Communications in Contemporary Mathematics", 28 pages, extended and revised version of the original "Homogeneous 2nd order PDEs on the adjoint contact manifold of a simple complex Lie group"

References in corpus (1)

• Exceptionally simple PDE

Cited by in corpus (2)

• Exceptionally simple PDE
• Exceptionally simple super-PDE for $F(4)$