Compact symplectic group ($USp(2n)\cong Sp(n)$)

The subgroup $\{U\in U(2n):UJ{U}^T=J\}$, with $J=\left(\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right)$, is the compact symplectic group, often denoted $Sp(n)$ or $USp(2n)$. It preserves both a positive Hermitian metric and a nondegenerate complex alternating form.

The matrix logarithm near the identity maps $Sp(n)=\{A\in U(2n):AJ{A}^t=J\}$ onto an open set of its real compact symplectic Lie algebra $\{X:X^{\ast }=-X,XJ+J{X}^t=0\}$. Its inverse is the matrix exponential. These statements follow by applying the logarithm to $A^{\ast }=A^{-1}$ and $A^t=J^{-1}{A}^{-1}J$, and differentiating $e^{tX}J{e}^{t{X}^t}$. Left translations provide manifold charts at every group element. In block form $X=\left(\begin{array}{cc}P& Q\\ -\bar{Q}& \bar{P}\end{array}\right)$ with $P^{\ast }=-P$ and $Q^t=Q$, giving real dimension $n^2+n(n+1)=n(2n+1)$.