existence and uniqueness of compact real form

Let G be a semisimplePlanetmathPlanetmathPlanetmathPlanetmath complex Lie group. Then there exists a unique (up to isomorphismPlanetmathPlanetmathPlanetmathPlanetmath) real Lie groupMathworldPlanetmath K such that K is compactPlanetmathPlanetmath and a real form of G. Conversely, if K is compact, semisimple and real, it is the real form of a unique semisimple complex Lie group G. The group K can be realized as the set of fixed points of a special involutionPlanetmathPlanetmathPlanetmath of G, called the Cartan involution.

For example, the compact real form of SLn, the complex special linear groupMathworldPlanetmath, is SU(n), the special unitary group. Note that SLn is also a real form of SLn, but is not compact.

The compact real form of SOn, the complex special orthogonal groupMathworldPlanetmath, is SOn, the real orthogonal groupMathworldPlanetmath. SOn also has other, non-compact real forms, called the pseudo-orthogonal groups.

The compact real form of Sp2n, the complex symplectic group, is less well-known. It is (unfortunately) also usually denoted Sp(2n), and consists of n×nunitaryPlanetmathPlanetmathquaternion matrices, that is,


where M* denotes M conjugate transposeMathworldPlanetmath. This different from the real symplectic group Sp2n.

Title existence and uniqueness of compact real form
Canonical name ExistenceAndUniquenessOfCompactRealForm
Date of creation 2013-03-22 13:23:37
Last modified on 2013-03-22 13:23:37
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 5
Author bwebste (988)
Entry type Theorem
Classification msc 22E10