A Vogan diagram is a Dynkin diagram with an involution, and the vertices fixed by the involution may be painted. They represent real simple Lie algebras, and two diagrams are said to be equivalent if they represent the same Lie algebra. In this article we classify the equivalence classes of all Vogan diagrams. In doing so, we find that the underlying Dynkin diagrams have certain properties in graph painting. We show that this combinatorial property provides an easy classification for most of the simply-laced Dynkin diagrams.  2004 Published by Elsevier Inc.

Abstract. A Vogan diagram is a set of involution and painting on a Dynkin diagram. It selects a real form, or equivalently an involution, from a complex simple Lie algebra. We introduce the double Vogan diagram, which is two sets of Vogan diagrams superimposed on an affine Dynkin diagram. They correspond to pairs of commuting involutions on complex simple Lie algebras, and therefore provide an independent classification of the simple locally symmetric pairs.

An extended Vogan diagram is an extended Dynkin diagram with a diagram involution, such that the vertices fixed by the involution can be painted or unpainted. Every extended Vogan diagram represents an almost compact real form of some affine Kac-Moody Lie algebra. Two diagrams may represent isomorphic algebras, and in this case we say that the diagrams are equivalent. In this paper, we classify the equivalence classes of extended Vogan diagrams, and provide a complete list of all diagrams within each class. It gives a combinatorial classification of the isomorphic classes of almost compact real forms of the affine Kac-Moody Lie algebras.

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