Homogeneous Freudenthal algebras and the first Tits construction

TL;DR

This paper investigates the conditions under which Freudenthal algebras are homogeneous, linking their structure to the first Tits construction and exploring their properties over various fields, including local and global cases.

Contribution

It provides necessary and sufficient conditions for homogeneity of Freudenthal algebras and connects these conditions with the first Tits construction, including new local-global principles.

Findings

Characterization of homogeneity conditions for Freudenthal algebras

Connection between homogeneity and the first Tits construction

Development of a local version of the Tits construction and a local-global principle

Abstract

Freudenthal algebras over a field are basically the same as Jordan algebras of degree $3$ remaining simple under all base field extensions. These algebras are intimately linked, via their automorphism groups and structure groups, to simple algebraic groups over arbitrary fields. Our main concern here will be the question of when these algebras are homogeneous in the sense that all their Jordan isotopes are isomorphic. We answer this question by presenting various necessary and sufficient conditions for homogeneity and by connecting it with the first Tits construction of cubic Jordan algebras, most notably through investigating Freudenthal division algebras over complete fields under a discrete valuation. We also study the first Tits construction in its own right by producing a local version of it, deriving a local-global principle, and by connecting it with the embeddibility of certain rank-$2$-tori into the automorphism group scheme of an Albert division algebra.