= Logarithm charts for the special unitary group
{title2=$\mathfrak{su}(n)=\{X:X^*=-X,\ \operatorname{tr}X=0\}$}
Near the identity, the <matrix logarithm> of a unitary matrix is skew-Hermitian. Restrict the logarithm's operator norm below $\pi/n$; determinant one then forces its trace to be zero, since $\det U=\exp(\operatorname{tr}\log U)$ and no nonzero multiple of $2\pi i$ fits this bound. The <matrix exponential> gives the inverse chart on a ball in the displayed real <Lie algebra>, of dimension $n^2-1$. Left translation supplies compatible charts on the whole <special unitary group>.
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