Compact symplectic group

Example

Let $n \ge 1$ and let $\mathbb{H}$ be the quaternions. For a quaternionic matrix $A \in M_n(\mathbb{H})$, let $A^*$ be its quaternionic conjugate transpose. The compact symplectic group is $$\mathrm{Sp}(n) = \set{A \in M_n(\mathbb{H}) : A^* A = I_n},$$ the group of $\mathbb{H}$-linear maps of $\mathbb{H}^n$ that preserve the quaternionic inner product $\ip{x}{y} = \sum_k \bar{x}_k y_k$. It is also called the quaternionic unitary group.

For $n = 1$, the condition reads $\bar{q} q = 1$, so $\mathrm{Sp}(1)$ is the group of unit quaternions, which is isomorphic to SU(2).

Writing $\mathbb{H}^n = \C^{2n}$ turns $\mathrm{Sp}(n)$ into a group of complex $2n \times 2n$ matrices, and then $$\mathrm{Sp}(n) = \mathrm{Sp}(2n, \C) \cap U(2n),$$ the intersection of the complex symplectic group with the unitary group. This is a closed and bounded subgroup of $GL_{2n}(\C)$, so $\mathrm{Sp}(n)$ is a compact Lie group.