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What's an example of a Lie group $G$ and matrix $X$ such that $e^X \in G$ but $x \notin \mathfrak{g}$, where $\mathfrak{g}$ is the associated Lie algebra?

This is the same as problem 2.10 in Bryan Hall's "Lie Groups, Lie Algebras, and Representations: An Elementary Introduction."

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    Take $G$ to be trivial...2012-10-05

1 Answers 1

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Take $H = \{ \pm I\}$ that has trivial Lie algebra because it is a finite group. The matrix

$X = \left(\begin{array}{cc} 0 & -\pi \\ \pi & 0 \end{array}\right)$

is such that $e^X = \left(\begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array}\right)$ but $X$ is not zero and hence is not in the Lie algebra.

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    @kjetil: take the direct product with any nontrivial Lie algebra / Lie group.2012-10-05