Journal: Communications in Contemporary Mathematics Year: 2017 Links:arXiv version

Abstract:

For each simple Lie algebra $\mathfrak{g}$ (excluding, for trivial reasons, type $C$) we find the lowest possible degree
of an invariant 2nd 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-Ampere equations have degree 1). For $\mathfrak{g}$ of type $A$ or $G_2$ we show that this gives all
invariant 2nd order PDEs. For $\mathfrak{g}$ of type $B$ and $D$
we provide an explicit formula for the lowest-degree invariant 2nd order PDEs. For $\mathfrak{g}$ of type $E$ and $F_4$ we prove uniqueness of the lowest-degree invariant 2nd order PDE; we also conjecture that uniqueness holds in type $D$.