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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.
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).
Last revised on January 14, 2023 at 17:35:54. See the history of this page for a list of all contributions to it.