Real form (Lie theory)

In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra g₀ is called a real form of a complex Lie algebra g if g is the complexification of g₀:

The notion of a real form can also be defined for complex Lie groups. Real forms of complex semisimple Lie groups and Lie algebras have been completely classified by Élie Cartan.

Real forms for Lie groups and algebraic groups

Using the Lie correspondence between Lie groups and Lie algebras, the notion of a real form can be defined for Lie groups. In the case of linear algebraic groups, the notions of complexification and real form have a natural description in the language of algebraic geometry.

Classification

Just as complex semisimple Lie algebras are classified by Dynkin diagrams, the real forms of a semisimple Lie algebra are classified by Satake diagrams, which are obtained from the Dynkin diagram of the complex form by labeling some vertices black (filled), and connecting some other vertices in pairs by arrows, according to certain rules.