A construction of Lie algebras from a class of ternary algebras

Abstract

A class of algebras with a ternary composition and alternating bilinear form is defined. The construction of a Lie algebra from a member of this class is given, and the Lie algebra is shown to be simple if the form is nondegenerate. A characterization of the Lie algebras so constructed in terms of their structure as modules for the three-dimensional simple Lie algebra is obtained in the case the base ring contains 1/2. Finally, some of the Lie algebras are identified; in particular, Lie algebras of type E8 are obtained. A construction of Lie algebras from Jordan algebras discovered independently by J. Tits [7] and M. Koecher [4] has been useful in the study of both kinds of algebras. In this paper, we give a similar construction of Lie algebras from a ternary algebra with a skew bilinear form satisfying certain axioms. These ternary algebras are a variation on the Freudenthal triple systems considered in [1]. Most of the results we obtain for our construction are parallel to those for the TitsKoecher construction (see [3, Chapter VIII]). In ?1, we define the ternary algebras, derive some basic results about them, and give two examples of such algebras. In ?2, the Lie algebras are constructed and shown to be simple if and only if the skew bilinear form is nondegenerate. In ?3, we give a characterization, in the case the base ring contains 1/2, of the Lie algebras obtained by our construction in terms of their structure as modules for the threedimensional simple Lie algebra. Finally, in ?4, we identify some of the simple Lie algebras obtained by our construction from the examples of ?1. In particular, we show that we can construct a Lie algebra of type E8 from a 56-dimensional space which is a module for a Lie algebra of type E7. A similar construction was given by H. Freudenthal in [2]. 1. A class of ternary algebras. We shall be interested in a module TM over an arbitrary commutative associative ring D with 1 which possesses an alternating bilinear form and a ternary product which satisfy (TI) = + z for x,y, ze M; (T2) = + x for x, y, z e 9; (T3) , w> = , z> + for x, y, z, w e 9; Received by the editors March 12, 1970 and, in revised form, June 19, 1970. AMS 1969 subject classifications. Primary 1730.