Invariants of Real Forms of Affine Kac-Moody Lie Algebras

Abstract

The theory of Vogan diagrams, which are Dynkin diagrams with an overlay of certain additional information, allows one to give a rapid classification of finite-dimensional real semisimple Lie algebras and to make use of this classification in practice. This paper develops a corresponding theory of Vogan diagrams for “almost compact” real forms of indecomposable nontwisted affine Kac–Moody Lie algebras. In this case also, the equivalence classes of Vogan diagrams correspond to the isomorphism classes of almost compact real forms. Although the real forms of such algebras had already been classified, the theory of Vogan diagrams introduces invariants for such algebras and makes it possible to locate a given real form within the classification.