Extended Vogan diagrams

Abstract

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.

Double Vogan diagrams and irreducible pseudo-Hermitian symmetric spaces

“…Underline vertex α to denote the operation F α of (2.2). We have (1, 5) → (1, 2, 5) → (2), so (1, 5) is equivalent to (2). Since (2) belongs to (3.1)(b), it follows that (1, 5) does not represent a pseudo-Hermitian symmetric pair.…”

Double Vogan diagrams and semisimple symmetric spaces

“…This can be regarded as the affine version of Theorem 2.1, or can be seen from explicit algorithms [6]. The condition r black m α = 2 on D r leads to three possibilities: They correspond precisely to the following types of g. Let z denote the center of k.…”

“…Corollary 1.4 says that there may be two extensions to inv(g c ), and so we need to check if the two extensions are conjugate. The method to judge equivalent diagrams was originally introduced in [1], [2], and is used to classify all the equivalence classes of Vogan diagrams [5] and extended Vogan diagrams [6]. We shall use a similar method here.…”

“…Here (6) is inner equivalent to (4), and (3) is inner equivalent to (7) [5]. There is no particular preference, so we omit (6) and (3) and let S be the following six diagrams:…”

“…By [6, Table 1] on E 1 8 , we see that (1, 2) cannot be transformed to (2) or (7) by any sequence of F α , even when we allow F 1 in [6]. So it remains only to show that (1, 2) is not equivalent to (5).…”