Albert algebras over and other rings

Abstract

AbstractAlbert algebras, a specific kind of Jordan algebra, are naturally distinguished objects among commutative nonassociative algebras and also arise naturally in the context of simple affine group schemes of type$\mathsf {F}_4$,$\mathsf {E}_6$, or$\mathsf {E}_7$. We study these objects over an arbitrary base ringR, with particular attention to the case$R = \mathbb {Z}$. We prove in this generality results previously in the literature in the special case whereRis a field of characteristic different from 2 and 3.