# 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 $\mathrm{SL}_{n}\mathbb{C}$, 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 $\mathrm{SL}_{n}\mathbb{C}$, but is not compact.

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

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

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

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 ExistenceAndUniquenessOfCompactRealForm 2013-03-22 13:23:37 2013-03-22 13:23:37 bwebste (988) bwebste (988) 5 bwebste (988) Theorem msc 22E10