nLab compact symplectic group

The compact symplectic groups

The compact symplectic groups


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


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