nLab compact symplectic group

Definition

The compact symplectic group (in $n$ dimensions) is the simply-connected, maximal compact real Lie group $Sp(n)$ sitting inside the complex symplectic group $Sp(2n,\mathbb{C})$. It can also be seen as the orthogonal group of $\mathbb{H}^n$. It may also be defined as the intersection of $Sp(2n,\mathbb{C})$ with the unitary group $\mathrm{U}(2n) = \mathrm{U}(2n,\mathbb{C})$ within the general linear group $GL(2n,\mathbb{C})$. It should not be confused with the real symplectic group $Sp(2n,\mathbb{R})$ (which is not simply-connected).

The compact symplectic group is also called the quaternionic unitary group, and see there for more.

Examples

• The group $Sp(1)$ is isomorphic to the special unitary group $SU(2)$ and also to the spin group $Spin(3)$. Its underlying smooth manifold is the 3-sphere.

• $Sp(2)$ is isomorphic to Spin(5) (see there for a proof).