Albert algebras over and other rings

Abstract

Albert 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 ring R, with particular attention to the case $R = \mathbb {Z}$ . 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.

A Generalization of the First Tits Construction

“…Finally, since one of x 2 , y 2 , z 2 is equal to zero, T A (µ −1 (x 2 ♯y 2 )z 2 ) = 0 = T A (x 2 µ −1 (y 2 ♯z 2 )). Therefore, applying these equalities and using (6), we deduce that ( 27) and (28) are equal, so T(x × y, z) = T(x, y × z).…”

“…Moreover, their automorphism group is an exceptional algebraic group of type F 4 , and their cubic norms have isometry groups of type E 6 . For some recent developments, see [2][3][4][5][6].…”