Satake diagram

In the mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram whose configurations classify simple Lie algebras over the field of real numbers. Properly, the Satake diagrams associated to a Dynkin diagram classify real forms of the complex Lie algebra corresponding to the Dynkin diagram.

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.

Examples

• 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).

References

• Bump, Daniel (2004), Lie groups, Graduate Texts in Mathematics, 225, Berlin, New York: Springer-Verlag, ISBN 978-0-387-21154-1, MR2062813 
• 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, MR1834454 
• Onishchik, A. L.; Vinberg, Ėrnest Borisovich (1994), Lie groups and Lie algebras III: structure of Lie groups and Lie algebras