Rational orbits of the space of pairs of exceptional Jordan algebras

TL;DR

This paper classifies generic rational orbits in a prehomogeneous vector space associated with pairs of exceptional Jordan algebras, linking them to algebraic structures like isotopes and cubic étale subalgebras, especially over split octonions.

Contribution

It establishes a bijective correspondence between rational orbits and algebraic structures such as isotopes and cubic étale subalgebras, extending understanding of orbit classification in exceptional Jordan algebra contexts.

Findings

Generic rational orbits correspond to pairs of isotopes and cubic étale subalgebras.

When octonions are split, orbits correspond to separable extensions of degree up to 3.

The classification provides a clear algebraic description of orbit structure in the prehomogeneous vector space.

Abstract

Let $k$ be a field of characteristic not equal to $2,3$, $\mathbb{O}$ an octonion over $k$ and $\mathcal{J}$ the exceptional Jordan algebra defined by $\mathbb{O}$. We consider the prehomogeneous vector space $(G,V)$ where $G=GE_6\times \mathrm{GL}(2)$ and $V=\mathcal{J}\oplus \mathcal{J}$. We prove that generic rational orbits of this prehomogeneous vector space are in bijective correspondence with $k$-isomorphism classes of pairs $(\mathcal{M},\mathfrak{n})$ where $\mathcal{M}$'s are isotopes of $\mathcal{J}$ and $\mathfrak{n}$'s are cubic \'etale subalgebras of $\mathcal{M}$. Also we prove that if $\mathbb{O}$ is split, then generic rational orbits are in bijective correspondence with isomorphism classes of separable extensions of $k$ of degrees up to $3$.