Real form (Lie theory)

The term real form is used in mathematics to that set defined over the real and complex numbers, objects, in particular algebraic structures related to each other. It is used mainly in the theory of Lie algebras and Lie groups.

A real Lie algebra is a real form of a complex Lie algebra if the complexification of is so

General it can be analogous to a real form of a complex vector space defined by the condition. A complex vector space has an infinite number of real forms, for example, or real forms of.

A real form of a complex Lie group is a subgroup whose Lie algebra is a real form of the Lie algebra of the complex Lie group.

Semi- Simple Lie algebras

Each semisimple complex Lie algebra has at least two real forms.

The one of the two real forms is a compact Lie algebra, ie the Killing form is negative definite.

The other of the two forms is a real cleavable Lie algebra, that there is a Cartan subalgebra, such that for all the adjoint mapping is diagonalizable.

In general, have any real forms.

Examples

The following list outlines a semisimple complex Lie algebra only the compact, then the fissile real form.

• (the Lie algebra of the compact symplectic group)

Classification

Real forms of a semisimple complex Lie algebra are classified by Satake diagrams, certain refinements of the Dynkin diagram of.

Representation theory

The complex representations of the corresponding 1:1 complex representations of: obtain all representations of by restricting the representations of the complexified Lie algebra. For example, the representation theory of equivalent to the representation theory of sl (2, C).