Symmetric Lie algebras

Abstract

Lie Rootsystems R are introduced, with axioms which reflect properties of the rootset of a Lie algebra L as structured by representations of compatible simple restricted rank 1 subquotients of L. The rank 1 Lie rootsystems and the rank 2 Lie rootsystems defined over Z p are classified up to isomorphism. Base, closure and core are discussed. The rootsystems of collapse on passage from R to Core R are shown to be of type S m . Given any Lie rootsystem R, its independent root pairs are shown to fall into eleven classes. Where the eleventh (anomoly) pair T 2 never occurs, it is shown that R is contained in R 0 + S (not always equal), where R 0 is a Witt rootsystem and S is a classical rootsystem. This result is of major importance to two papers (D. J. Winter, Generalized classical-Albert-Zassenhaus Lie algebras, to appear; Rootsystems of simple Lie algebras, to appear), since it implies that the rootsystems of the simple nonclassical Lie algebras considered there are Witt rootsystems. Toral Lie algebras and symmetric Lie algebras are introduced and studied as generalizations of classical-Albert-Zassenhaus Lie algebras. It is shown that their rootsystems are Lie rootsystems. The cores of toral Lie algebras are shown to be classical-Albert-Zassenhaus Lie algebras. These results form the basis for the abovementioned papers on rootsystems of simple Lie algebras and the classification of the rootsystems of two larger classes of Lie algebras, the generalized classical-Albert-Zassenhaus Lie algebras and the classical-Albert-Zassenhaus-Kaplansky Lie algebras. Symmetric Lie algebras are introduced as generalizations of classical-Albert-Zassenhaus Lie algebras. It is shown that their rootsets R are Lie rootsystems. Consequently, symmetric Lie algebras can be studied locally using the classification of rank 2 Lie rootsystems. This is done in detail for toral Lie algebras.