Classification of real forms up to inner automorphisms

Question

I hope to know the classification of real forms of complex simple Lie algebras of types $A$, $D$, $E$ up to inner automorphisms.

Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be real forms of a complex simple Lie algebra $\mathfrak{g}$. We say that they are equivalent if there is an isomorphism $\mathfrak{g}_1 \to \mathfrak{g}_2$ which extends to an inner automorphism of $\mathfrak{g}$. However, there may be real forms which are isomorphic but not equivalent.

Where can we find the classification of such equivalence classes of real forms?

Answer 1

The one case (for $\frak g$ simple) where the two definitions of real forms (up to $Aut(\frak g)$ versus $Int(\frak g)$) don't agree is the following.

There is a real form of $\frak g=\frak s\mathfrak o(2n,\mathbb C)$ denoted $\frak s\frak o^*(2n)$.

For $n\ge4$ even $\frak g$ has two subalgebras isomorphic to $\frak s\frak o^*(2n)$ which are related by $Aut(\frak g)$, but not by $Int(\frak g)$. If $n\ge 3$ is odd there is only one such algebra up to $Int(\frak g)$. The difference is because the center of the simply connected group is $\mathbb Z/2\times\mathbb Z/2$ if $n$ is even, and $\mathbb Z/4$ if $n$ is odd.

$n=4$ is particularly interesting: the triality automorphism of $D_4$ interchanges these two copies of $\mathfrak s\mathfrak o*(2n)$, as well as $\frak s\frak o(6,2)\simeq \frak s\frak o^*(2n)$.

Answer 2

I'd recommend looking at the article "Strong real forms and the Kac classification", by Jeffrey Adams -- it's an expository paper from 2005 which answers your question.

For more details, Adams is careful to distinguish between three notions which he calls "traditional real forms", "real forms", and "strong real forms". He works at the level of groups rather than Lie algebras, but let me repeat his definitions here.

Let $\mathfrak{g}$ be a simple complex Lie algebra. Let $G$ be the associated adjoint algebraic group over the complex numbers, so we identify $\mathfrak{g} = Lie(G)$ and $Int(\mathfrak{g}) = Int(G)$ and the homomorphism $G \rightarrow Int(G)$ is an isomorphism. There is a well-known short exact sequence: $$1 \rightarrow Int(G) \rightarrow Aut(G) \rightarrow Out(G) \rightarrow 1.$$ Adams defines:

• A traditional real form of $G$ is an equivalence class of involutions in $Aut(G)$, where equivalence is given by conjugation by $Aut(G)$: $\iota \sim \alpha \iota \alpha^{-1}$ for any $\alpha \in Aut(G)$.

• A real form of $G$ is an equivalence class of involutions in $Aut(G)$, where equivalence is given by conjugation by $Int(G)$: $\iota \sim Int(g) \iota Int(g^{-1})$ for all $g \in G$.

Associated to an involution $\iota$, there is a real algebraic group $G_R$, whose complexification coincides with $G$, and which has a maximal compact subgroup $K_R$ with complexification $K_C = G^\iota$.

Now, for the Lie algebra, let's make the analogous definitions:

• A traditional real form of $\mathfrak{g}$ is an equivalence class of involutions in $Aut(\mathfrak{g})$, where equivalence is given by conjugation by $Aut(\mathfrak{g})$.

• A real form of $\mathfrak{g}$ is an equivalence class of involutions in $Aut(\mathfrak{g})$, where equivalence is given by conjugation in $Int(\mathfrak{g})$.

Since $G$ is chosen to be the adjoint group, $Aut(G) = Aut(\mathfrak{g})$ and $Int(G) = Int(\mathfrak{g})$. When $\iota$ is an involution in $Aut(\mathfrak{g})$, there exists an antiholomorphic involution $\theta$ of $\mathfrak{g}$, such that $\mathfrak{g}^\theta$ is a compact real form, and $\iota \theta = \theta \iota$. In this way, $\iota$ yields a real Lie algebra $\mathfrak{g}^{\iota \theta} = \mathfrak{g}^{\theta \iota}$.

Furthermore, such a $\theta$ is unique up to $Int(\mathfrak{g}^\iota )$, so for any $\theta' = \gamma \theta \gamma^{-1}$ another such involution, we find that $$\mathfrak{g}^{\iota \theta'} = \mathfrak{g}^{\iota \gamma \theta \gamma^{-1}} = \mathfrak{g}^{\gamma \iota \theta \gamma^{-1}} = \gamma \left( \mathfrak{g}^{\iota \theta} \right).$$

If we consider an inner conjugate of $\iota$: $\iota' = \delta \iota \delta^{-1}$ for $\delta \in Int(\mathfrak{g})$, then one may choose a commuting Cartan involution $\theta' = \delta \theta \delta^{-1}$. In this way, we find $$\mathfrak{g}^{\iota' \theta'} = \delta \left( \mathfrak{g}^{\iota \theta} \right).$$

What this boils down to is that the suggested notion of inner-equivalence classes of real forms of $\mathfrak{g}$ coincides precisely with Adams notion of a real form of $G$, or my adaptation above to a real form of $\mathfrak{g}$. Hence the answer is given by classifying the equivalence classes of involutions in $Aut(G)$, where equivalence is given by conjugation in $Int(G)$.

Now, to come to the Kac classification that Adams discusses. We refer back to Adams for the details (or Adams-Barbasch-Vogan, Chapter 2).

Fix $G$, and also an involution in $\gamma \in Out(G)$; this fixes an "inner class" of real forms. Let $c$ be the order of $\gamma$: $c \in \{ 1,2 \}$. Let $\tilde \Delta_\gamma$ denote the affine twisted Dynkin diagram associated to the pair $(\mathfrak{g}, \gamma)$, with vertices $\{ \alpha_0, \ldots, \alpha_\ell \}$. (Fold the usual Dynkin diagram of $\mathfrak{g}$ by the automorphism $c$, and extend it by the lowest root). Let $n_i$ denote the usualy numbering of this Dynkin diagram, based on multiplicities in the highest root, for example.

A Kac marking of the Dynkin diagram will mean a subset $S \subset \{0, \ldots, n \}$ of the vertices of $\tilde \Delta_\gamma$, such that $$\sum_{i \in S} n_i = 2/c.$$ Thus at most two vertices are marked, and $n_i \leq 2$ for all marked vertices $i \in S$.

$Z(G_{sc})$ acts on $\tilde \Delta_\gamma$, and the orbits on the set of Kac markings parameterize the real forms within the inner class given by $\gamma$.

I think this is the best you'll get -- there aren't too many cases, but I don't know where to find a table of them.