## Dmitri V. Alekseevsky - J. Gutt - G. Manno - G. Moreno

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

created by moreno on 03 Oct 2017

modified on 15 Sep 2020

[

BibTeX]

*Published Paper*

**Inserted:** 3 oct 2017

**Last Updated:** 15 sep 2020

**Journal:** Communications in Contemporary Mathematics

**Year:** 2019

**Links:**
arXiv version

**Abstract:**

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

**Tags:**
MSC2014-GEOGRAL

**Keywords:**
simple Lie algebras, adjoint variety, Lagrangian Grassmannian, second order PDEs, symmetries of PDEs, invariant theory