The structure of cyclic Lie algebras

Abstract

Simple toral rank 1 Lie algebras have been classified in Wilson [8]. This paper is concerned with the structure of a nonsimple toral rank 1 Lie algebra with respect to a specified "toral rank 1" Cartan subalgebra or, equivalently, with the structure of a nonsimple graded Lie algebra where the grading is the cyclic group grading determined by a specific "toral rank 1" Cartan subalgebra. Such graded Lie algebras are called cyclic Lie algebras, to distinguish them from ungraded toral rank 1 Lie algebras and from graded toral rank 1 Lie algebras where the grading is not a cyclic group grading determined by a "toral rank 1" Cartan subalgebra. The structure theorems on cyclic Lie algebras of this paper are established by studying L L in terms of its graded subalgebras and quotient algebras. Their importance is due to the central role which cyclic Lie algebras play in the theory of Lie algebra rootsystems.