Groups of type $\mathrm{E}_6$ and $\mathrm{E}_7$ over Rings via Brown Algebras and Related Torsors

Abstract

We study structurable algebras and their associated Freudenthal triple systems over commutative rings. The automorphism groups of these triple systems are exceptional groups of type E 7 , and we realize groups of type E 6 as centralizers. When 6 is invertible, we further give a geometric description of homogeneous spaces of type E 7 /E 6 , and show that they parametrize principal isotopes of Brown algebras. As opposed to the situation over fields or local rings, we show that such isotopes may be non-isomorphic.

Albert algebras over and other rings

“…That is, if 𝑄(𝐽) 𝑄 (𝐽 ), then J and 𝐽 are isotopic. The paper [Als22] provides an example related to this construction over rings.…”