Complexification - Knowlpedia

Let $V$ be a real vector space. Its complexification is the complex vector space

V_{\mathbb C}:=V\otimes_{\mathbb R}\mathbb C,

formed as an algebraic tensor product by extending scalars from the real numbers to the complex numbers.

Canonical embedding and coordinates

The canonical real-linear map $V\to V_{\mathbb C}$ sends $v$ to $v\otimes1$ . Every element has a unique expression

v\otimes1+w\otimes i, \qquad v,w\in V,

usually abbreviated $v+iw$ . If $V$ is finite-dimensional, then

\dim_{\mathbb C}V_{\mathbb C}=\dim_{\mathbb R}V.

Universal property

For every complex vector space $W$ , restriction along $V\to V_{\mathbb C}$ gives a natural bijection

\operatorname{Hom}_{\mathbb C}(V_{\mathbb C},W) \cong \operatorname{Hom}_{\mathbb R}(V,W),

where $W$ on the right is regarded as a real vector space. Explicitly, a real-linear map $f:V\to W$ extends uniquely to the complex-linear map

f_{\mathbb C}(v\otimes z)=z\,f(v).

This characterizes complexification independently of a basis.

Conjugation and real form

Complexification carries a canonical conjugation

\overline{v\otimes z}=v\otimes\overline z.

Its fixed-point subspace is the embedded copy of $V$ . Conversely, a complex vector space equipped with an antilinear involution is the complexification of its fixed real subspace.

Any real-linear map $T:V\to W$ complexifies to $T_{\mathbb C}:V_{\mathbb C}\to W_{\mathbb C}$ , defined by $T_{\mathbb C}(v\otimes z)=T(v)\otimes z$ . This construction preserves compositions, direct sums, tensor products, kernels, and images.

Remarks

One must distinguish complexifying a real vector space from merely forgetting or restoring scalar structure. If $V$ is already complex and $V_{\mathbb R}$ denotes its underlying real vector space, then

(V_{\mathbb R})_{\mathbb C}\cong V\oplus\overline V,

not just $V$ . The two summands correspond to the $+i$ and $-i$ eigenspaces of the complexified original complex structure.

References

Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: Section 1.2, complexification and complex-linear decompositions.