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 Satake (1960, p. 109) and were originally used to classify real simple Lie algebras. Additionally, they also classify symmetric pairs ${\displaystyle ({\mathfrak {g}},{\mathfrak {k}})}$ of Lie algebras, where ${\displaystyle {\mathfrak {g}}}$ is semisimple.

More concretely, given a complex semisimple Lie algebra ${\displaystyle {\mathfrak {g}}}$, the Satake diagrams made from ${\displaystyle {\mathfrak {g}}}$'s Dynkin diagram classify the involutions of ${\displaystyle {\mathfrak {g}}}$'s root system that extend to an anti-linear involutive automorphism σ of ${\displaystyle {\mathfrak {g}}}$. The fixed points ${\displaystyle {\mathfrak {g}}^{\sigma }}$ are then a real form of ${\displaystyle {\mathfrak {g}}}$. The same Satake diagrams also classify the involutions of ${\displaystyle {\mathfrak {g}}}$'s root system that extend to a (linear) involutive automorphism σ of ${\displaystyle {\mathfrak {g}}}$. The fixed points ${\displaystyle {\mathfrak {k}}}$ form a complex Lie subalgebra of ${\displaystyle {\mathfrak {g}}}$, so that ${\displaystyle ({\mathfrak {g}},{\mathfrak {k}})}$ 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 Tits (1966), 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.