Determine differentiable homomorhpisms from $SO_2$ to $GL_n$.
Are the answers the maps $t\rightarrow e^{tA}$,where $A\in M_n$ and $e^{2\pi A}=I_n$? Or is there any deeper description on $A$? Thank you
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1$\begingroup$ Are you familiar with the representation theory of the given group? $\endgroup$– Tobias KildetoftMay 25, 2020 at 17:48
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$\begingroup$ Yes, i know a little @Tobias Kildetoft $\endgroup$– ysTuanMay 25, 2020 at 17:54
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1$\begingroup$ Well, this question is essentially asking you to name the $n$-dimensional differentiable representations of it. $\endgroup$– Tobias KildetoftMay 25, 2020 at 17:57
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$\begingroup$ To build on Tobias' comment: Do you mean the group scheme $SO_2$ over $\operatorname{Spec}\mathbb{R}$ or do you mean the group of points $SO_2(\mathbb{R})$? Because if you mean the latter it may be easier (conceptually) to use the isomorphism $SO_2(\mathbb{R}) \cong \mathbb{S}^{1}$ and look for $n$-dimensional differentiable representations of the unit circle. $\endgroup$– GeoffMay 25, 2020 at 18:32
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