Compact symplectic group

For $n\geq 1$ , regard $\mathbb H^n$ as a right vector space over the quaternion division algebra with Hermitian form

\langle v,w\rangle=\sum_{r=1}^n\overline{v_r}w_r.

The compact symplectic group is

\operatorname{Sp}(n) =\{A\in\operatorname{GL}(n,\mathbb H):A^*A=I\}.

Thus $\operatorname{Sp}(n)$ consists precisely of the right $\mathbb H$ -linear isometries of $\mathbb H^n$ , with group operation given by composition. As a closed, bounded real matrix group, it is a compact Lie group.

Complex matrix realization

Writing each quaternionic matrix as a complex matrix of twice the size identifies

\operatorname{Sp}(n) \cong \operatorname{Sp}(n,\mathbb C)\cap\operatorname{U}(2n).

Here $\operatorname{U}(2n)$ preserves the standard Hermitian form, while the complex symplectic factor preserves the standard nondegenerate alternating complex bilinear form.

Structure and low-rank example

The group $\operatorname{Sp}(n)$ is connected and simply connected. Its real Lie algebra consists of quaternionic matrices $X$ satisfying $X^*+X=0$ , and has dimension $n(2n+1)$ . For $n=1$ , it is the group of unit quaternions; hence $\operatorname{Sp}(1)$ is isomorphic to $\operatorname{SU}(2)$ , with underlying manifold $S^3$ .

Conventions and scope

References

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Progress in Mathematics 140, Birkhäuser, 2002. Author-hosted text. Relevant: §I.17, especially (1.135), Proposition 1.136, and Proposition 1.139.