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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... – Qiaochu Yuan Oct 5 '12 at 1:14
up vote 2 down vote accepted

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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(For what it's worth, Maple says that $e^X = -I$, but you can just change $H$ to accommodate it. In order to see why your answer must be wrong, note that the nontrivial element of $H$ has determinant $-1$, so is in a different connected componenet of $GL_2$ than $I$. In particular, it cannot be $e^X$ for any $X$). – Jason DeVito Oct 5 '12 at 2:16
@JasonDeVito I have edited my answer. I messed up before :D – user38268 Oct 5 '12 at 3:02
Thanks so much! I didn't realize what a simple question this is until I read your clear answer. – Pinkwater Oct 5 '12 at 3:09
¿But is there any example with a matrix Lie group which is not trivial as in this example (that is, not a finite grpup, and with a Lie algebra of dimension larger than zero? – kjetil b halvorsen Oct 5 '12 at 5:26
@kjetil: take the direct product with any nontrivial Lie algebra / Lie group. – Qiaochu Yuan Oct 5 '12 at 6:19

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