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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?

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2 Answers 2

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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, 2017 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 $V$. $\endgroup$ May 22, 2017 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, 2021 at 3:32
  • $\begingroup$ @JohnBaez: What's an example for this, and am I right in my hope that it involes non-semisimple Lie algebras, i.e. for semisimple LAs there is an equivalence? This question also quotes a proposition that we at least have a one-to-one correpsondence on irreducible representations: math.stackexchange.com/q/1408894/96384. $\endgroup$ Nov 18, 2021 at 17:01
  • $\begingroup$ The proposition in that question asserts an equivalence between complex irreducible representations of a real Lie algebra $\mathfrak{g}$ and complex irreducible representations of $\mathfrak{g}_{\mathbb{C}}$. I was claiming nonisomorphic real representations of $\mathfrak{g}$ can, upon complexification, give isomorphic complex representations of $\mathfrak{g}_{\mathbb{C}}$. So, what are you actually interested in? $\endgroup$
    – John Baez
    Nov 18, 2021 at 22:10
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The comment section to the other answer brought up an issue which maybe deserves another answer.

What is true:

For a real Lie algebra $\mathfrak g$, there is a one-to-one correspondence between

  1. representations of $\mathfrak g$ on complex vector spaces; that is, $\mathbb R$-linear Lie algebra homomorphisms $\mathfrak g \rightarrow \mathrm{End}_{\mathbb C}(V)$ for a complex vector space $V$, and
  2. representations of the complexification $\mathfrak g_\mathbb C := \mathbb C \otimes_{\mathbb R} \mathfrak g$ on complex vector spaces; that is, $\mathbb C$-linear Lie algebra homomorphisms $\mathfrak g_\mathbb C \rightarrow \mathrm{End}_{\mathbb C}(V)$ for a complex vector space $V$,

Namely, the quote from Hall's book in the OP explicitly states that $\mathfrak h$ is a complex Lie algebra, and Hall and José Carlos Santos' answer apply this to the case $\mathfrak h = \mathrm{End}_{\mathbb C}(V)$ for a complex vector space $V$.

This equivalence is kind of common knowledge and comes up regularly on this site. Cf. In what sense are complex representations of a real Lie algebra and complex representations of the complexified Lie algebra equivalent?, https://math.stackexchange.com/a/3258221/96384, Bijection between the complex representations of a real Lie algebra and the complex representations of its complexification, Obtaining representation of a real Lie algebra from the complexification "by restriction".

Where one has to be careful:

If in the title and the question body the phrase "real representation of $\mathfrak g$" is supposed to mean a representation of $\mathfrak g$ on a real vector space $W$, i.e. a Lie algebra homomorphism $\mathfrak g \rightarrow \mathrm{End}_{\mathbb R}(W)$, then

  1. such a representation also does define a complex representation of $\mathfrak g_\mathbb C$, but that needs an extra step absent from the construction in the quote from Hall;
  2. it is not true that this process is "reversible" in general; indeed, not only can it happen that non-isomorphic real representations of $\mathfrak g$ give out isomorphic complex representations of $\mathfrak g_\mathbb C$ (as John Baez pointed out), but more strikingly, not every complex representation of $\mathfrak g_{\mathbb C}$ comes from a real representation of $\mathfrak g$ via such procedure.

Ad 1: The extra step is that we first have to complexify the real vector space $W$ to $W_\mathbb C := W \otimes_\mathbb R \mathbb C$, and extend our real representation $\mathfrak g \rightarrow \mathrm{End}_{\mathbb R}(W)$ via the natural $\mathbb R$-linear inclusion $\mathrm{End}_{\mathbb R}(W) \hookrightarrow \mathrm{End}_{\mathbb C}(W_\mathbb C)$. Only then do we have a map $\mathfrak g \rightarrow \mathrm{End}_{\mathbb C}(V)$ (namely, $V := W_\mathbb C$) as in what was true above, and can proceed like there.

Ad 2: But it is precisely this extra step which is not reversible. Note also that in José Carlos Santos' answer, what comes out is a complex representation of $\mathfrak g$, i.e. a map $\mathfrak g \rightarrow \mathrm{End}_{\mathbb C}(V)$ with a complex vector space $V$. There is no reason, and in general it is not true, that that complex vector space $V$ has a real subspace $W$ which is stable under the Lie algebra action and whose complex span is all of $W$.

As a standard example, consider the obvious complex representation of the real Lie algebra $\mathfrak{su}_2$ on $V:=\mathbb C^2$ via writing $\mathfrak{su}_2$ as the matrices $\pmatrix{ai & b+ci\\-b+ci &-ai}$ with $a,b,c \in \mathbb R$. If there were a two-(real-)dimensional subspace $W$ of $V$ stable under the $\mathfrak{su}_2$-action, we would have a nonzero map $\mathfrak{su}_2 \rightarrow \mathfrak{gl}_\mathbb R(W) \simeq \mathfrak{gl}_2(\mathbb R)$. It's easily seen that such a map would induce an isomorphism $\mathfrak{su}_2 \simeq \mathfrak{sl}_2(\mathbb R)$, which is absurd for a plethora and more of reasons.

It is, rather, an interesting question to decide for a given real Lie algebra $\mathfrak g$, which of its complex representations "come" from real representations in such way, which ones are "truly complex", and as it turns out there is a third case, namely they can come from "quaternionic" representations. Cf. part A of https://math.stackexchange.com/a/4026224/96384 as well as What property of the root system means a Lie algebra has complex structure?, or the discussion in https://math.stackexchange.com/a/3712110/96384.

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