Exercise 15, § 9

Lie Groups and Lie Algebras, III, 400

(a) Let $x \in \mathfrak{sl}(2, \mathbf{R})$ and $\Delta = \det x$ . Show that

e^x = \mathrm{ch}\, \sqrt{-\Delta}.I + \frac{\mathrm{sh}\, \sqrt{-\Delta}}{\sqrt{-\Delta}}.x \quad \text{if } \Delta < 0

e^x = \cos \sqrt{\Delta}.I + \frac{\sin \sqrt{\Delta}}{\sqrt{\Delta}}.x \qquad \text{if } \Delta > 0.

(b) Let $H = \mathbf{SL}(2, \mathbf{R})$ . Show that $g = \begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{pmatrix} \in H$ is the image of $\exp_H$ if and only if $\lambda > 0$ or $\lambda = -1$ . If $\lambda = 0$ , all the one-parameter subgroups containing $g$ are equal.

Solution

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