The structure of cyclic Lie algebras

Winter, David J.

doi:10.1090/S0002-9939-1987-0884453-4

The structure of cyclic Lie algebras
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by David J. Winter

Proc. Amer. Math. Soc. 100 (1987), 213-219

DOI: https://doi.org/10.1090/S0002-9939-1987-0884453-4

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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 $L$ 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.

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