[[Lie algebra]]
# Complexification of a real Lie algebra

Let $\mathfrak{u}$ be a [[Lie algebra]] over $\mathbb{R}$.
The **complexification** $\mathfrak{g}$ of $\mathfrak{u}$ is the [[complexification]] of $\mathfrak{g}$ as a vector space with a natural bracket, #m/def/lie 
namely
$$
\begin{align*}
\mathfrak{g} = \mathbb{C} \otimes_{\mathbb{R}} \mathfrak{u} = \mathfrak{u} \oplus i\mathfrak{u}
\end{align*}
$$
and
$$
\begin{align*}
[z \otimes x, w \otimes y] = zw \otimes [x,y]
\end{align*}
$$
for $z,w \in \mathbb{C}$ and $x,y \in \mathfrak{u}$.
Hence $\mathfrak{g}$ is the [[tensor product of a Lie algebra and a commutative algebra]] as well as an [[induced module]].


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#state/tidy | #lang/en | #SemBr