On rational orbits in some prehomogeneous vector spaces

TL;DR

This paper classifies rational orbits in certain prehomogeneous vector spaces over fields with characteristic not 2, linking these orbits to composition algebras, Freudenthal algebras, and involutions on division algebras.

Contribution

It provides a parametrization of rational orbits in specific prehomogeneous vector spaces and relates these to algebraic structures like composition and Freudenthal algebras, extending understanding of their classifications.

Findings

Parametrization of composition algebras over field k.

Description of reduced Freudenthal algebras of dimensions 6 and 9.

Classification of involutions of the second kind on central division algebras.

Abstract

Let $k$ be a field with characteristic different from $2$. In this paper, we describe the $k$-rational orbit spaces in some irreducible prehomogeneous vector spaces $(G,V)$ over $k$, where $G$ is a connected reductive algebraic group defined over $k$ and $V$ is an irreducible rational representation of $G$ with a Zariski dense open orbit. We parametrize all composition algebras over the field $k$ in terms of the orbits in some of these representations. This leads to a parametric description of the reduced Freudenthal algebras of dimensions $6$ and $9$ over $k$ (if $\text{char}(k)\neq 2,3$). We also get a parametrization for the involutions of the second kind defined on a central division $K$-algebra $B$ with center $K$, a quadratic extension of the underlying field $k$.