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Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
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Root space decomposition
What is the root system for the special unitary lie algebra $\mathfrak{su}(p, q)$. Remind that these are matrices of the form
$\left(
\begin{array}{cc}
X & Y \\
\overline{Y}^t & Z \\
\end …