Albert algebras over Z and other rings

TL;DR

This paper investigates Albert algebras, special Jordan algebras linked to exceptional groups, over arbitrary rings including integers, extending known results from fields to more general base rings.

Contribution

It generalizes existing results on Albert algebras from fields to arbitrary rings, especially the integers, broadening their algebraic understanding.

Findings

Established properties of Albert algebras over arbitrary rings

Extended classification results from fields to rings like Z

Connected Albert algebras to simple affine group schemes

Abstract

Albert algebras, a specific kind of Jordan algebra, are naturally distinguished objects among commutative non-associative algebras and also arise naturally in the context of simple affine group schemes of type $F_4$, $E_6$, or $E_7$. We study these objects over an arbitrary base ring $R$, with particular attention to the case of the integers. We prove in this generality results previously in the literature in the special case where $R$ is a field of characteristic different from 2 and 3.