See also compact symplectic group.
This Lie group is the analog of the unitary group as one passes from the complex numbers to the quaternions.
The quaternionic unitary group $Sp(n)$ is the group of quaternion-unitary transformations of $\mathbb{H}^n$. It is also called the compact symplectic group, since both it and the symplectic group $Sp(2n, \mathbb{R})$ are real forms of the complex Lie group $Sp(2n,\mathbb{C})$, and it is the compact form.
$Sp(1) \simeq$ Spin(3) (this Prop.)
$Sp(2) \simeq$ Spin(5) (this Prop.)
A Riemannian manifold of dimension $4n$ is called a quaternion-Kähler manifold if its holonomy group is a subgroup of the quotient group Sp(n).Sp(1) of the direct product group $Sp(n) \times Sp(1)$. If it is even a subgroup of just the $Sp(n)$ factor, then it is called a hyperkähler manifold.
Last revised on January 18, 2023 at 21:26:23. See the history of this page for a list of all contributions to it.