# existence and uniqueness of compact real form

Let $G$ be a semisimple^{} complex Lie group. Then there exists a unique (up to isomorphism^{}) real Lie group^{} $K$ such that $K$ is compact^{} 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 involution^{} of $G$, called the Cartan involution.

For example, the compact real form of ${\mathrm{SL}}_{n}\u2102$, the complex special linear group^{}, is $\mathrm{SU}(n)$, the special unitary group. Note that ${\mathrm{SL}}_{n}\mathbb{R}$ is also a real form of ${\mathrm{SL}}_{n}\u2102$, but is not compact.

The compact real form of ${\mathrm{SO}}_{n}\u2102$, the complex special orthogonal group^{}, is ${\mathrm{SO}}_{n}\mathbb{R}$, the real orthogonal group^{}. ${\mathrm{SO}}_{n}\u2102$ also has other, non-compact real forms, called the pseudo-orthogonal groups.

The compact real form of ${\mathrm{Sp}}_{2n}\u2102$, the complex symplectic group, is less well-known. It is (unfortunately) also usually denoted $\mathrm{Sp}(2n)$, and consists of $n\times n$ “unitary^{}” quaternion matrices, that is,

$$\mathrm{Sp}(2n)=\{M\in {\mathrm{GL}}_{n}\mathbb{H}|M{M}^{*}=I\}$$ |

where ${M}^{*}$ denotes $M$ conjugate transpose^{}. This different from the real symplectic group ${\mathrm{Sp}}_{2n}\mathbb{R}$.

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 |