Symplectic Group

Overview

If we generalize the idea of Special Orthogonal Group (Rotation) on ${\mathbb{R}}^n$ to ${\mathbb{H}}^n$, the group of linear transformations preserving the inner product is the symplectic group $Sp(n)$

The inner product defined in ${\mathbb{H}}^n$ is formally the same as the inner product defined on ${\mathbb{C}}^n$ (see Overview)

$$Sp(n)=\{A\in M_n(\mathbb{H}):A^{\ast }A=I\}$$

Avoid use of quaternions

Since it’s difficult to define determinant for quaternion matrices (${\mathbb{H}}^n$ is not a vector space over $\mathbb{H}$), we avoid this problem by thinking $n\times n$ quaternion matrices as $2n\times 2n$ complex matrices

Example

$Sp(1)$ consists of the $1\times 1$ quaternion matrices, which preserves length in $\mathbb{H}$, so they are simply the group of unit quaternions. Therefore, we have $Sp(1)\cong SU(2)$