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I know that $\mathrm{GL}_{n}(\mathbb{C})$ is not simply connected. Therefore I don’t quite understand the correspondence between representations of $\mathrm{GL}_{n}(\mathbb{C})$ and $\mathfrak{gl}_{n}(\mathbb{C})$. Given a representation of $\mathrm{GL}_{n}(\mathbb{C})$ how can i construct a representation of $\mathfrak{gl}_{n}(\mathbb{C})$? As I understand in the other direction it could be constructed via exponential map.

Also I would be grateful if someone could recommend some literature on the same topic for different realizations of $\mathfrak{gl}_{\infty}(\mathbb{C})$.

Matthew Willow
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  • If $\rho:GL_n(\Bbb C)\rightarrow GL(V)$ is a representation, then its differential $d\rho:\mathfrak{gl}_n(\Bbb C)\rightarrow \mathfrak{gl}(V)$ is a Lie algebra representation. See the answer by José Carlos Santos here. The problem only is that this process will not give all Lie algebra representations for $\mathfrak{gl}_n(\Bbb C)$, but you only wanted to "construct a representation. – Dietrich Burde Jul 04 '23 at 13:16
  • @Deitrich_Brude Thanks – Matthew Willow Jul 24 '23 at 22:36

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