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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The components of the automorphism group of a Jordan algebra

Author: S. Robert Gordon
Journal: Trans. Amer. Math. Soc. 153 (1971), 1-52
MSC: Primary 17.40
MathSciNet review: 0286854
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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.

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Keywords: Semisimple Jordan algebra, characteristic zero, automorphism group, algebraic components, fixed points, derivation algebra, Lie algebra, Koecher-Tits algebra, root space
Article copyright: © Copyright 1971 American Mathematical Society