# Does any complex representation of $\mathfrak{g}_\mathbb{C}$ define a real representation of $\mathfrak{g}$?

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.
• 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 $V$. – José Carlos Santos May 22 '17 at 18:32
• 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.) – John Baez Feb 28 at 3:32