A canonical isomorphism of Lie algebras

Question

Let $g$ be a Lie algebra/$\mathbb{C}$. I would like to investigate the existence of a canonical Lie algebra isomorphism/$\mathbb{C}$ of the form $g\otimes_{\mathbb{R}}\mathbb{C}\rightarrow g\oplus g$. (Here we consider $g$ on the left as a Lie algebra/$\mathbb{R}$ and then take the extension of scalars.) By considering examples I am led to think there should be such an isomorphism. However, the obvious isomorphism of vector spaces $x\otimes z\mapsto(\operatorname{Re} z \ x,\operatorname{Im} z \ x)$ does not seem to be a Lie algebra isomorphism. What mistake am I doing/what is the correct isomorphism/is the result in question even true in this generality?

Answer 1

This is not quite right. The correct statement is that for $V$ a complex vector space, there is a natural isomorphism

$$V \otimes_{\mathbb{R}} \mathbb{C} \cong V \oplus \overline{V}$$

where $\overline{V}$ denotes the conjugate vector space; this has a modified scalar multiplication, namely $c \cdot v = \bar{c} v$. Because this isomorphism is natural, it continues to apply to vector spaces with extra structure, such as Lie algebras, but also such as commutative and associative algebras, and even vector bundles.

For Lie algebras we get

$$\mathfrak{g} \otimes_{\mathbb{R}} \mathbb{C} \cong \mathfrak{g} \oplus \overline{\mathfrak{g}}$$

and it is not in general true that $\mathfrak{g} \cong \overline{\mathfrak{g}}$, although counterexamples are hard to come by because this isomorphism holds for any $\mathfrak{g}$ with a real form, and all of the usual examples (including all complex semisimple Lie algebras) have real forms.

This isomorphism can be obtained as follows. Write

$$V \otimes_{\mathbb{R}} \mathbb{C} \cong V \otimes_{\mathbb{C}} \mathbb{C} \otimes_{\mathbb{R}} \mathbb{C}.$$

This observation reduces the problem to trying to understand $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C}$ as a $(\mathbb{C}, \mathbb{C})$-bimodule. As a bimodule it decomposes as a direct sum

$$\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \cong \mathbb{C} \oplus \overline{\mathbb{C}}$$

where the first $\mathbb{C}$ has the usual bimodule structure (so tensoring with it is the identity) and the second $\mathbb{C}$ has bimodule structure twisted by complex conjugation on one side (so tensoring with it is the functor $\overline{V}$).

It's an exercise from here to write down an explicit formula for the isomorphism above, which will give an explicit formula for the desired isomorphism.