Isomorphism classes of $k$-involutions of algebraic groups of type $E_6$

Abstract

Automorphisms of order $2$ are studied in order to understand generalized symmetric spaces. The groups of type $E_6$ we consider here can be realized as both the group of linear maps that leave a certain determinant invariant, and also as the identity component of the automorphism group of a class of structurable algebras known as Brown algebras. We will classify the $k$-involutions of these groups of type $E_6$ using aspects of both descriptions, and give detailed descriptions of representatives over certain fields including algebraically closed fields, $\mathbb{R}$, $\mathbb{F}_p$, and $\mathbb{Q}_p$.