In the mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by Satake (1960, p.109) whose configurations classify simple Lie algebras over the field of real numbers. The Satake diagrams associated to a Dynkin diagram classify real forms of the complex Lie algebra corresponding to the Dynkin diagram.
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 not the same as Vogan diagrams of a Lie group, although they look similar.
A Satake diagram is obtained from a Dynkin diagram by blackening some vertices, and connecting other vertices in pairs by arrows, according to certain rules.
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(K) 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.
|This section requires expansion. (December 2009)|
- Compact Lie algebras correspond to the Satake diagram with all vertices blackened.
- A table can be found at (Onishchik & Vinberg 1994, Table 4, pp. 229–230).
- Bump, Daniel (2004), Lie groups, Graduate Texts in Mathematics 225, Berlin, New York: Springer-Verlag, ISBN 978-0-387-21154-1, MR 2062813
- Helgason, Sigurdur (2001), Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics 34, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2848-9, MR 1834454
- Onishchik, A. L.; Vinberg, Ėrnest Borisovich (1994), Lie groups and Lie algebras III: structure of Lie groups and Lie algebras
- Satake, Ichirô (1960), "On representations and compactifications of symmetric Riemannian spaces", Annals of Mathematics. Second Series 71: 77–110, ISSN 0003-486X, MR 0118775
- Satake, Ichiro (1971), Classification theory of semi-simple algebraic groups, Lecture Notes in Pure and Applied Mathematics 3, New York: Marcel Dekker Inc., ISBN 978-0-8247-1607-3, MR 0316588
- Spindel, Philippe; Persson, Daniel; Henneaux, Marc (2008), "Spacelike Singularities and Hidden Symmetries of Gravity", Living Reviews in Relativity 11 (1)
- Tits, Jacques (1966), "Classification of algebraic semisimple groups", Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Providence, R.I.: American Mathematical Society, pp. 33–62, MR 0224710
- Tits, Jacques (1971), "Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque", Journal für die reine und angewandte Mathematik 247: 196–220, doi:10.1515/crll.1971.247.196, ISSN 0075-4102, MR 0277536