Complexification of a real Lie algebra

Core idea

For a real Lie algebra $\mathfrak h$ , its complexification is

\mathfrak h_{\mathbb C}:=\mathfrak h\otimes_{\mathbb R}\mathbb C,

with the unique complex-bilinear bracket satisfying

[X\otimes z,Y\otimes w]=[X,Y]\otimes zw.

The map $X\mapsto X\otimes1$ identifies $\mathfrak h$ with a real Lie subalgebra, and $\dim_{\mathbb C}\mathfrak h_{\mathbb C}=\dim_{\mathbb R}\mathfrak h$ .

Universal property and conjugation

Every real-linear Lie algebra homomorphism from $\mathfrak h$ to the underlying real algebra of a complex Lie algebra extends uniquely to a complex-linear homomorphism from $\mathfrak h_{\mathbb C}$ . Complex conjugation on the second tensor factor is an antilinear involution whose fixed Lie algebra is $\mathfrak h$ .

Complexification must not be confused with forgetting complex scalars. If $\mathfrak g$ is complex, then

(\mathfrak g_{\mathbb R})_{\mathbb C}\cong\mathfrak g\oplus\overline{\mathfrak g},

not generally $\mathfrak g$ alone.

References

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002, §I.3. Publisher record.
Jean-Pierre Serre, Complex Semisimple Lie Algebras, Springer, 1987, Chapter I. Publisher record.