S4-symmetry on the Tits Construction of Exceptional Lie Algebras and Superalgebras

The classical Tits construction provides models of the exceptional simple Lie algebras in terms of a unital composition algebra and a degree three simple Jordan algebra. A couple of actions of the symmetric group S4 on this construction are given. By means of these actions, the models provided by the Tits construction are related to models of the exceptional Lie algebras obtained from two different types of structurable algebras. Some models of exceptional Lie superalgebras are discussed too. Introduction In a previous paper [EO06], the authors have studied those Lie algebras with an action of the symmetric group of degree 4, denoted by S4, by automorphisms. Under some conditions, these Lie algebras are coordinatized by the structurable algebras introduced by Allison [All78]. The purpose of this paper is to show how a structurable algebra of an admissible triple appears naturally when considering an action by automorphisms of the symmetric group S4 on the classical Tits construction of the exceptional Lie algebras [Tit66]. This can be extended to the superalgebra setting. A different S4 action will be considered too, related to the structurable algebras consisting of a tensor product of two composition algebras. This provides connections of the Tits construction to other models of the exceptional Lie algebras. The paper is structured as follows. The first section will be devoted to show how the symmetric group S4 acts by automorphisms of the split Cayley algebra. Sections 2 and 3 will review, respectively, the classical Tits construction [Tit66] of the exceptional Lie algebras, and the structurable algebras of admissible triples attached to separable Jordan algebras of degree 3. Then Section 4 will show how to extend the action of S4 on the Cayley algebra to an action by automorphisms on the Tits Construction. The associated coordinate algebra will be shown to be isomorphic to the structurable algebra attached to the Jordan algebra used in the construction. The proof involves many computations, but the isomorphism given is quite natural. Section 5 will extend the results of the previous section to the superalgebra Date: October 20, 2006.