Vogan diagram

Assume the Axiom of Choice (The Axiom of Choice).

Let ${\mathfrak{g}}_0$ be a finite-dimensional real semisimple Lie algebra with Cartan involution $\theta$, complexification $\mathfrak{g}$ and Cartan involution data as in Cayley transform of a theta-stable Cartan subalgebra. Let ${\mathfrak{h}}_0$ be a maximally compact $\theta$-stable Cartan subalgebra (Theta-stable Cartan subalgebras and their compact and split parts), with complexification $\mathfrak{h}$ and root system $\mathrm{\Phi }$. By Cayley transforms connect theta-stable Cartans in the classification the Cartan subalgebra ${\mathfrak{h}}_0$ has no real roots, so every root $\alpha \in \mathrm{\Phi }$ is imaginary or complex.

Compatible positive system. A positive system ${\mathrm{\Delta }}^{+}$ of $\mathrm{\Phi }$ (Positive systems and simple roots) is compatible with $({\mathfrak{h}}_0,\theta )$ if it is $\theta$-stable, $\theta ({\mathrm{\Delta }}^{+})={\mathrm{\Delta }}^{+}$; equivalently, if its base $\mathrm{\Delta }$ satisfies $\theta (\mathrm{\Delta })=\mathrm{\Delta }$, where $\theta \alpha =\alpha \circ {\theta }^{-1}$ on ${\mathfrak{h}}^{\ast }$. Compatible positive systems exist. Work on the real vector space ${\mathfrak{h}}_{\mathbb{R}}=i{\mathfrak{t}}_0\oplus {\mathfrak{a}}_0$, on which $B$ is positive definite and roots are real-valued, as proved in steps 1.1 and 2.1 of Cayley transforms connect theta-stable Cartans in the classification. Since there are no real roots, each root restricts nontrivially to $i{\mathfrak{t}}_0$. Choose $H\in i{\mathfrak{t}}_0$ outside their finitely many kernels (A finite-dimensional vector space over an infinite field is not a finite union of proper subspaces), and take ${\mathrm{\Delta }}^{+}=\{\alpha :\alpha (H)>0\}$. This is a positive system under the Killing identification with the real root space. Because $\theta H=H$, it is theta-stable. A theta-stable positive system has a theta-stable base because theta preserves decompositions into sums of positive roots; conversely a theta-stable base determines a theta-stable positive system by the nonnegative simple-root expansions (Simple roots form a signed integral basis). When the root set is empty, the positive system and base are empty and $H=0$ works.

The Vogan diagram of a triple. Let ${\mathrm{\Delta }}^{+}$ be a compatible positive system with base $\mathrm{\Delta }$. The Vogan diagram of the triple $({\mathfrak{g}}_0,{\mathfrak{h}}_0,{\mathrm{\Delta }}^{+})$ is the Dynkin diagram of $\mathrm{\Phi }$ relative to $\mathrm{\Delta }$ (Dynkin diagram with edge multiplicity and arrow convention) together with the following additional structure:

• the involution induced by $\theta$ on $\mathrm{\Delta }$: since $\theta ({\mathrm{\Delta }}^{+})={\mathrm{\Delta }}^{+}$, $\theta$ permutes the simple roots, and $\alpha \mapsto \theta \alpha$ is an involutive automorphism of the Dynkin diagram; its orbits on $\mathrm{\Delta }$ have one or two elements, and the two-element orbits are marked (equivalently the two vertices of such an orbit are joined by the standard pair-labelling);
• each $\theta$-fixed simple root $\alpha \in {\mathrm{\Delta }}^{\theta }$ is painted if the root $\alpha$ is noncompact, that is ${\mathfrak{g}}_{\alpha }\subseteq \mathfrak{p}$, and is left unpainted if $\alpha$ is compact, that is ${\mathfrak{g}}_{\alpha }\subseteq \mathfrak{k}$ (Cayley transform of a theta-stable Cartan subalgebra).

Abstract Vogan diagrams. An abstract Vogan diagram is a finite-type Dynkin diagram, including its edge multiplicities and arrows, equipped with an automorphism $\tau$ satisfying ${\tau }^2=\mathrm{i}\mathrm{d}$ and a subset $P$ of its fixed vertices. The vertices of $P$ are painted; fixed vertices outside $P$ are unpainted. Two-element orbits are marked as pairs, and their vertices are not painted. Disconnected diagrams and permutations of isomorphic components are allowed; the empty diagram is allowed for the zero algebra.

Equivalently, over a based reduced crystallographic root system $(\mathrm{\Phi },\mathrm{\Delta })$, these data are a base-preserving root-system involution $\tau$ and a subset $P\subseteq {\mathrm{\Delta }}^{\tau }$. A root-system isomorphism here means a linear bijection carrying roots to roots and preserving the Cartan integers $2(\alpha ,\beta )\mathrm{/}(\beta ,\beta )$; no absolute choice of scale on individual irreducible components is part of the diagram.

For a realized triple one can additionally record the eigenvalue $\epsilon (\alpha )\in \{1,-1\}$ of $\theta$ on each imaginary root space. This is well defined by Root spaces of a complex semisimple Lie algebra are one-dimensional. These eigenvalues are properties of the realizing Lie-algebra involution; an independently assigned function on all fixed roots is not part of an abstract Vogan diagram. In particular, the rule $\epsilon (\alpha +\beta )=\epsilon (\alpha )\epsilon (\beta )$ for fixed roots alone is not a definition of such an extension. In type $A_2$ with the two simple roots interchanged, their sum is fixed although neither simple root is fixed, so that rule alone leaves its sign undetermined.

Equivalence. The equivalence relation on abstract Vogan diagrams is generated by the following reversible moves:

1. Relabel by an isomorphism of Dynkin diagrams preserving multiplicities and arrows, transporting the involution and the painted subset.
2. At a painted fixed simple vertex $\alpha$, perform the reflection/painting move $F_{\alpha }$. Keep the underlying diagram and its involution, keep $\alpha$ painted, and for each other fixed simple vertex $\beta$ reverse its color exactly when the Cartan integer $2(\beta ,\alpha )\mathrm{/}(\alpha ,\alpha )$ is odd. Vertices in two-element orbits remain unpainted.

Thus the second move reverses adjacent fixed colors, except at a longer root joined to $\alpha$ by a double edge. Nonadjacent colors do not change. The rule is independent of root-length scale and is involutive, since the same set of vertices is toggled twice and $\alpha$ stays painted. A chain of zero moves is allowed. These are the standard finite Vogan reflection/painting moves; the existence and classification theorems establish their relationship with real forms. This definition neither assumes a classification result nor adds arbitrary signs on nonsimple roots.