Generalized classical Albert-Zassenhaus Lie algebras

Abstract

Two very large classes GCAZ and CAZK of Lie algebras are introduced, which contain all sums of classical, Albert-Zassenhaus, generalized Witt algebras of Kaplansky and associated holomorphs. Their rootsystems R are classified up to isomorphism. The group Aut L of automorphisms of L is shown to contain extensions of the Weyl group of R and the inner automorphism groups of classical Lie algebra complements of the Witt subalgebra of L. The Weyl group extension in Aut L acts transitively by conjugation on the classical complements, under general conditions.