The components of the automorphism group of a Jordan algebra

Abstract

Let F \mathfrak {F} be a semisimple Jordan algebra over an algebraically closed field Φ \Phi of characteristic zero. Let G G be the automorphism group of F \mathfrak {F} and Γ \Gamma the structure groups of F \mathfrak {F} . General results on G G and Γ \Gamma are given, the proofs of which do not involve the use of the classification theory of simple Jordan algebras over Φ \Phi . Specifically, the algebraic components of the linear algebraic groups G G and Γ \Gamma are determined, and a formula for the number of components in each case is given. In the course of this investigation, certain Lie algebras and root spaces associated with F \mathfrak {F} are studied. For each component G i {G_i} of G G , the index of G G is defined to be the minimum dimension of the 1 1 -eigenspace of the automorphisms belonging to G i {G_i} . It is shown that the index of G i {G_i} is also the minimum dimension of the fixed-point spaces of automorphisms in G i {G_i} . An element of G G is called regular if the dimension of its 1 1 -eigenspace is equal to the index of the component to which it belongs. It is proven that an automorphism is regular if and only if its 1 1 -eigenspace is an associative subalgebra of F \mathfrak {F} . A formula for the index of each component G i {G_i} is given. In the Appendix, a new proof is given of the fact that the set of primitive idempotents of a simple Jordan algebra over Φ \Phi is an irreducible algebraic set.