Proposition 3.39 of Hall's Lie Groups, Lie Algebras And Representations:

"Let $\mathfrak{g}$ be a real Lie algebra, $\mathfrak{g}_\mathbb{C}$ its complexification, and $\mathfrak{h}$ an arbitrary complex Lie algebra. Then every real Lie algebra homomorphism of $\mathfrak{g}$ into $\mathfrak{h}$ extends uniquely to a complex Lie algebra homomorphism of $\mathfrak{g}_\mathbb{C}$ into $\mathfrak{h}$."

In particular this means that any real representation of $\mathfrak{g}$ defines a complex representation of $\mathfrak{g}_\mathbb{C}$.

Question: Does the converse hold? Does any complex representation of $\mathfrak{g}_\mathbb{C}$ define a real representation of $\mathfrak{g}$? Are there any conditions for when this may or may not hold?


Of course it does. If the Lie algebra $\mathfrak{g}_{\mathbb C}$ acts on a complex vector space $V$, simply consider the restriction of this action to $\mathfrak g$. To be more precise, if $X\in\mathfrak g$ and if $v\in V$, then define $X.v$ as $(X\otimes 1).v$. I am assuming here that Hall defined $\mathfrak{g}_{\mathbb C}$ as $\mathfrak{g}\bigotimes_{\mathbb{R}}\mathbb{C}$. If he used another definition, please say which one did he use.

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    $\begingroup$ So then there is a bijection between the real representations of the algebra and complex representations of the complexification? Or can,say, two complex representations restrict to the same real representation? $\endgroup$ – Okazaki May 22 '17 at 18:02
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    $\begingroup$ I would not use the word bijection in this context. But, yes, every representaion of $\mathfrak g$ on a complex vector space $V$ induces a representation of $\mathfrak{g}_{\mathbb C}$ on $V$ and every representation of $\mathfrak{g}_{\mathbb C}$ on $V$ is induced by some representation of $\mathfrak g$ on $V$. $\endgroup$ – José Carlos Santos May 22 '17 at 18:32
  • $\begingroup$ Nonisomorphic real representations of $\mathfrak{g}$ can, using the process Hall describes, give isomorphic complex representations of $\mathfrak{g}_{\mathbb{C}}$. So, this functor from real representations of $\mathfrak{g}$ to complex representations of $\mathfrak{g}_{\mathbb{C}}$ is not an equivalence of categories. ("Equivalence" is the right generalization of "bijection" in this context.) $\endgroup$ – John Baez Feb 28 at 3:32

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