The components of the automorphism group of a Jordan algebra

Abstract:

Let $\mathfrak {F}$ be a semisimple Jordan algebra over an algebraically closed field $\Phi$ of characteristic zero. Let $G$ be the automorphism group of $\mathfrak {F}$ and $\Gamma$ the structure groups of $\mathfrak {F}$. General results on $G$ and $\Gamma$ are given, the proofs of which do not involve the use of the classification theory of simple Jordan algebras over $\Phi$. Specifically, the algebraic components of the linear algebraic groups $G$ and $\Gamma$ 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 $\mathfrak {F}$ are studied. For each component ${G_i}$ of $G$, the index of $G$ is defined to be the minimum dimension of the $1$-eigenspace of the automorphisms belonging to ${G_i}$. It is shown that the index of ${G_i}$ is also the minimum dimension of the fixed-point spaces of automorphisms in ${G_i}$. An element of $G$ is called regular if the dimension of its $1$-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 $1$-eigenspace is an associative subalgebra of $\mathfrak {F}$. A formula for the index of each component ${G_i}$ 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 $\Phi$ is an irreducible algebraic set.