Some groups of Transformations defined by Jordan Algebras. III

Abstract

In I of this series of papers we introduced the notion of the generic minimum polynomial and generic norm and trace for finite dimensional strictly power associative algebras. We determined these functions for any central simple Jordan algebra J and we introduced the groups L( J) (the norm preserving or n. p. group of J) as the group of 1 - 1 linear transformations η of J onto J such that N(a ^η) = N(a), a in J, N the generic norm in J. Under a small restriction on the base field we showed that the group of automorphisms G( J) is the subgroup of L( J) characterized by the condition 1^η = 1. This result together with known theorems on automorphisms enabled us to determine the group L( J) for J central simple and special. In II of the present series we investigated the structure of the group of automorphisms of any reduced exceptional simple Jordan algebra J. The present paper is devoted to a similar study of the group L( J) for J a reduced exceptional Jordan algebra over any field of characteristic $\ne$ 2, 3. We shall refer to these groups as groups of type E _6I since for an algebraically closed base field of characteristic 0 the group L( J) is the Lie group E ₆ in the Killing-Cartan classification of simple Lie groups²).

This research was supported by the United States Air Force Office of Scientific Research and Development Command under Contract AF 49 (C 38) 515. Reproduction in whole or in part is permitted for any purpose of the United States Government.