Invariant Monge-Ampère equations on contactified para-Kähler manifolds

Abstract

We develop a method for describing invariant Monge-Ampère equations in the sense of V. Lychagin and T. Morimoto (MAE) on a homogeneous contact manifold $N$ of a semisimple Lie group $G$, which is the contactification of the homogeneous symplectic manifold $M=G/H={\mathrm{Ad}}_{G}Z\subset \mathfrak{g}$, where $M$ is the adjoint orbit of a splittable closed element $Z$ of the Lie algebra $\mathfrak{g}=\mathrm{Lie}(G)$. The method is then applied to a ten-dimensional semisimple orbit $M$ of the exceptional Lie group ${\mathsf{G}}_2$ and a complete list of mutually non-equivalent MAEs on $N$ is obtained.