= Logarithm charts for the compact symplectic group
{title2=$A\mapsto\log(A_0^{-1}A)$}
The <matrix logarithm> near the identity maps $Sp(n)=\{A\in U(2n):AJA^t=J\}$ onto an open set of its real <compact symplectic Lie algebra> $\{X:X^*=-X,\ XJ+JX^t=0\}$. Its inverse is the <matrix exponential>. These statements follow by applying the logarithm to $A^*=A^{-1}$ and $A^t=J^{-1}A^{-1}J$, and differentiating $e^{tX}Je^{tX^t}$. Left translations provide <manifold charts> at every group element. In block form $X=\left(\begin{smallmatrix}P&Q\\-\bar Q&\bar P\end{smallmatrix}\right)$ with $P^*=-P$ and $Q^t=Q$, giving real dimension $n^2+n(n+1)=n(2n+1)$.
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