Strictly regular elements in Freudenthal triple systems

Abstract

Strictly regular elements play a role in the structure theory of Freudenthal triple systems analogous to that played by idempotents in nonassociative algebras with identity. In this paper we study the coordinatization of reduced triple systems relative to a connected pair of strictly regular elements and use the explicit form of strictly regular elements in terms of the coordinatization to prove uniqueness of the coordinatizing Jordan algebra, as well as several generalizations of known results regarding groups of transformations related to triple systems. Finally, we classify forms of a particularly important triple system (the representation module for the Lie algebra E7) over finite, p-adic or real fields. Following Freudenthal [71, Meyberg [91 and Brown [11 have introduced an axiomatic approach to the study of the ternary algebraic structure of the minimal dimensional module for the lie algebra E7. In particular, this module is one of a class of ternary algebras we refer to as Freudenthal Triple Systems (FTS). In this paper we analyze further the internal structure of such algebras using as a basic tool the Peirce decomposition relative to a pair of supplementary strictly regular elements. Using this decomposition we obtain, in a manner similar to [1 ], a coordinatization theorem-every simple, reduced FTS is isomorphic to VQ), ?S a member of a small class of Jordan algebras specified in ?1. In ?6 we analyze the forms of strictly regular elements in K2?) and use the results to show I) lN(S) if and only if ?and A are norm equivalent. In ?7 the action of the group of q-similarities on the strictly regular elements is studied, yielding conjugacy theorems generalizing a result of Brown [1], as well as information regarding ratios of similitudes generalizing results of Faulkner [41 and Seligman (unpublished). Results of this section are useful also in the classification problem for Lie algebras of type E7. In the final section we classify completely all forms of lQ(?), 3 exceptional central simple, for finite, p-adic, or real fields. In [51, symplectic algebras (an asymmetric version of FTS's) are studied, yielding different proofs of several results obtained in this paper. Received by the editors December 22, 1971. AMS (MOS) subject classifications (1970). Primary 17A30; Secondary 17B60.