Some Groups of Transformations defined by Jordan Algebras. I

Abstract

It is well known that the Lie algebra of derivations of the exceptional Jordan algebra M 3 8 over an algebraically closed field of characteristic 0 or over the field of reals is the exceptional Lie algebra F 4 of Killing-Cartan and that the Lie algebra of linear transformations in M 3 8 leaving a certain cubic norm form defined on M 3 8 invariant is the exceptional Lie algebra E 6 (Chevalley-Schafer [7], Freudenthal [11]). The program envisaged in the present series of papers is the extension of these results and their group analogues to arbitrary central simple Jordan algebras. The present paper gives the definition of the groups and Lie algebras in the general case and gives their determination for special Jordan algebras with some restrictions on the base field. In a second paper we plan to consider the groups of automorphisms of arbitrary reduced exceptional simple Jordan algebras and in a third paper we hope to study the generalized groups of types E 6 which arise from these Jordan algebras.