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 of Lie algebras, where is semisimple.
More concretely, given a complex semisimple Lie algebra, the Satake diagrams made from 's Dynkin diagram classify the involutions of 's root system that extend to an anti-linear involutive automorphism of. The fixed points are then a real form of. The same Satake diagrams also classify the involutions of 's root system that extend to a involutive automorphism of. The fixed points form a complex Lie subalgebra of, so that 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 consists of a root system that spans and a linear involution of that satisfies.
Let be the set of roots fixed by and let
is called the restricted root system.
The Satake diagram of a σ-root system is obtained as follows: Let be simple roots of such that are simple roots of. We can define an involution of by having
The Satake diagram is then obtained from the Dynkin diagram describing by blackening the vertices corresponding to, and by drawing arrows between the white vertices that are interchanged by.

Satake diagram of a real semisimple Lie algebra

Let be a real semisimple Lie algebra and let be its complexification. Define the map
This is an anti-linear involutive automorphism of real Lie algebras and its fixed-point set is our original.
Let be a Cartan subalgebra that satisfies and is maximally split, i.e. when we split into -eigenspaces, the -eigenspace has maximal dimension. induces an anti-linear involution on :
If is a root vector, one can show that
. Consequently, preserves
the root system of. We thus obtain a σ-root system whose Satake diagram is the Satake diagram of.

Satake diagram of a symmetric pair

Let be a symmetric pair of complex Lie algebras where is semisimple, i.e. let
be an involutive Lie algebra automorphism of and let
be its fixed-point set. It is shown in that these symmetric pairs, 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 consists of a subset of finite type and a diagram automorphism satisfying

The permutation coincides with
For with, we have, where

Given an admissible pair, we can define a σ-root system by equipping the root system of with the involution

Classification of Satake diagrams

In it is proven that every Satake diagram arising from a real semisimple Lie algebra is a disconnected union of

two times the same Dynkin diagram, with arrows matching the vertices
one of the following diagrams:

Name	Diagram	Type of restricted root system	Example real semisimple Lie algebra	Example symmetric pair

Examples

Compact Lie algebras correspond to the Satake diagram with all vertices blackened. This corresponds to the symmetric pair
Split Lie algebras correspond to the Satake diagram with only white and unpaired vertices.
A table can be found at.

Differences between Satake and Vogan diagrams

Both Satake and Vogan diagrams are used to classify semisimple Lie groups or algebras over the reals and both consist of Dynkin diagrams enriched by blackening a subset of the nodes and connecting some pairs of vertices by arrows. Satake diagrams, however, can be generalized to any field and fall under the general paradigm of Galois cohomology, whereas Vogan diagrams are defined specifically over the reals. Generally speaking, the structure of a real semisimple Lie algebra is encoded in a more transparent way in its Satake diagram, but Vogan diagrams are simpler to classify.
The essential difference is that the Satake diagram of a real semisimple Lie algebra with Cartan involution θ and associated Cartan pair is defined by starting from a maximally noncompact θ-stable Cartan subalgebra, that is, one for which and is as small as possible, whereas Vogan diagrams are defined starting from a maximally compact θ-stable Cartan subalgebra, that is, one for which and is as large as possible.
The unadorned Dynkin diagram, when interpreted as a Satake diagram, represents the split real form of the Lie algebra, whereas it represents the compact form when interpreted as a Vogan diagram.

Generalisation: Satake—Tits diagrams

Suppose that G is an algebraic group defined over a field k, such as the reals. We let S be a maximal split torus in G, and take T to be a maximal torus containing S defined over the separable algebraic closure K of k. Then G has a Dynkin diagram with respect to some choice of positive roots of T. This Dynkin diagram has a natural action of the Galois group of K/''k. Also some of the simple roots vanish on S''. The Satake–Tits diagram is given by the Dynkin diagram D, together with the action of the Galois group, with the simple roots vanishing on S colored black. In the case when k is the field of real numbers, the absolute Galois group has order 2, and its action on D is represented by drawing conjugate points of the Dynkin diagram near each other, and the Satake–Tits diagram is called a Satake diagram.