Satake Diagram

In the mathematical study of Lie algebras and Lie groups, Satake diagrams are a generalization of Dynkin diagrams that classify involutions of root systems that are relevant in several contexts. They were introduced in and were originally used to classify real simple Lie algebras. Additionally, they also classify symmetric pairs $(\mathfrak,\mathfrak)$ of Lie algebras, where $\mathfrak$ is semisimple.

More concretely, given a complex semisimple Lie algebra $\mathfrak$, the Satake diagrams made from $\mathfrak$'s Dynkin diagram classify the involutions of $\mathfrak$'s root system that extend to an anti-linear involutive automorphism of $\mathfrak$. The fixed points $\mathfrak^\sigma$ are then a real form of $\mathfrak$. The same Satake diagrams also classify the involutions of $\mathfrak$'s root system that extend to a (linear) involutive automorphism of $\mathfrak$. The fixed points $\mathfrak$ form a complex Lie subalgebra of $\mathfrak$, so that $(\mathfrak,\mathfrak)$ is a symmetric pair.

More generally, the Tits index or Satake–Tits diagram of a reductive algebraic group over a field is a generalization of the Satake diagram to arbitrary fields, introduced by , that reduces the classification of reductive algebraic groups to that of anisotropic reductive algebraic groups.

Satake diagrams are distinct from Vogan diagrams although they look similar.

Definition

Let be a real vector space. A σ-root system $(R,\sigma)$ consists of a root system $R\subset V$ that spans and a linear involution of that satisfies $\sigma(R)=R$.

Let $R_\bullet\subset R$ be the set of roots fixed by and let
$\Sigma := \left\.$ is called the restricted root system.

The Satake diagram of a σ-root system $(R,\sigma)$ is obtained as follows: Let $\alpha_1,\dots,\alpha_n$ be simple roots of such that $\alpha_,\dots,\alpha_n$ are simple roots of $R_\bullet$. We can define an involution of $\$ by having
$\sigma(\alpha_i) = \alpha_ + \Z R_\bullet\qquad (i=1,\dots,n-p).$ The Satake diagram is then obtained from the Dynkin diagram describing by blackening the vertices corresponding to $\alpha_,\dots,\alpha_n$, and by drawing arrows between the white vertices that are interchanged by .

Satake diagram of a real semisimple Lie algebra

Let $\mathfrak_\R$ be a real semisimple Lie algebra and let $\mathfrak=\mathfrak_\R\otimes\C$ be its complexification. Define the map
$\sigma:\mathfrak\to\mathfrak,\qquad X\otimes z\mapsto X\otimes\overline.$
This is an anti-linear involutive automorphism of real Lie algebras and its fixed-point set is our original $\mathfrak_\R$.

Let $\mathfrak\le\mathfrak$ be a Cartan subalgebra that satisfies $\sigma(\mathfrak)=\mathfrak$ and is maximally split, i.e. when we split $\mathfrak$ into -eigenspaces, the $-1$-eigenspace has maximal dimension. induces an anti-linear involution on $\mathfrak^*$:
$\sigma^*(\lambda)(v) = \overline\qquad (\lambda\in\mathfrak^*,v\in\mathfrak).$
If $X\in\mathfrak_\alpha$ is a root vector, one can show that
$\sigma(X)\in\mathfrak_$. Consequently, preserves
the root system of $\mathfrak$. We thus obtain a σ-root system $(R,\sigma^*)$ whose Satake diagram is the Satake diagram of $\mathfrak_\R$.

Satake diagram of a symmetric pair

Let $(\mathfrak,\mathfrak)$ be a symmetric pair of complex Lie algebras where $\mathfrak$ is semisimple, i.e. let
be an involutive Lie algebra automorphism of $\mathfrak$ and let
$\mathfrak$ be its fixed-point set. It is shown in that these symmetric pairs (even for $\mathfrak$ an infinite-dimensional Kac-Moody algebra), or equivalently these involutive automorphisms, can be classified using
so-called admissible pairs. These admissible pairs describe again a σ-root system that can be obtained from the automorphism , and the Satake diagrams that arise this way are exactly the ones listed in and the Satake diagrams obtained by blackening all vertices.

Definition
Given a Dynkin diagram with vertex set , an admissible pair $(I_\bullet, \tau)$ consists of a subset $I_bullet$ of finite type and a diagram automorphism satisfying
* $\tau^2=\operatorname$
* $\tau(I_\bullet)=I_\bullet$
* The permutation $\tau, _$ coincides with $-w_\bullet$ (where $w_\bullet$ is the longest element of the Weyl group generated by the vertices in $I_\bullet$)
* For $j\in I\setminus I_\bullet$ with $\tau(i)=i$, we have $\alpha_j(\rho^\vee_\bullet)\in\Z$, where $\rho^\vee_\bullet = \frac\sum_ \alpha^\vee.$

Given an admissible pair $(I_\bullet,\tau)$, we can define a σ-root system by equipping the root system of with the involution $\sigma = -w_\bullet\circ\tau$

Classification of Satake diagrams

In it is proven that every Satake diagram arising from a real semisimple Lie algebra (equivalently: symmetric pair $(\mathfrak,\mathfrak)$ with $\mathfrak$ semisimple) is a disconnected union of
* two times the same Dynkin diagram (with white vertices), with arrows matching the vertices
* one of the following diagrams:

Examples

* Compact Lie algebras correspond to the Satake diagram with all vertices blackened. This corresponds to the symmetric pair $(\mathfrak,\mathfrak)$
* Split Lie algebra