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I am trying to get a grasp on what a representation is, and a professor gave me a simple example of representing the group $Z_{12}$ as the twelve roots of unity, or corresponding $2\times 2$ matrices. Now I am wondering how $\operatorname{GL}(1,\mathbb{C})$ and $\operatorname{GL}(2,\mathbb{R})$ are related, since the elements of both groups are automorphisms of the complex numbers. $\operatorname{GL}(\mathbb{C})$, the group of automorphisms of C, is (to my understanding) isomorphic to both $\operatorname{GL}(1,\mathbb{C})$ and $\operatorname{GL}(2,\mathbb{R})$ since the complex numbers are a two-dimensional vector space over $\mathbb{R}$. But it doesn't seem like these two groups are isomorphic to each other.

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    An easy way to see that these are distinct is to note that they have different dimensions: $\dim_\mathbb{C}(\mathfrak{gl}_1(\mathbb{C}))=1$, while $\dim_\mathbb{R}(\mathfrak{gl}_2(\mathbb{R}))=4$.2010-11-30
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    @Aaron: if the OP knew how to compute dimensions of Lie algebras, presumably he/she wouldn't be asking this question.2010-11-30
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    Okay. Sorry if that was rude or anything. I know very little about representation theory, and I definitely didn't know that people study representations of Lie groups totally independently of Lie algebras.2010-12-01

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