\sigma:{C}\to{C}
\sigma(z)={\barz}
{C}
{C}={R} ⊕ i{R}
The conjugation map is antilinear:
\sigma(λz)={\barλ}\sigma(z)
\sigma(z1+z2)=\sigma(z1)+\sigma(z2)
\sigma:V\toV
VR\subsetV
VR ⊗ R{C}\toV
is an isomorphism. Conversely any vector space that is the complexificationof a real vector space has a natural real structure.
One first notes that every complex space V has a realification obtained by taking the same vectors as in the original set and restricting the scalars to be real. If
t\inV
t ≠ 0
t
it
\dimRV=2\dimCV
\sigma:V\toV
\sigma\circ\sigma=idV
v\inV
{v=v++v-
v+={1\over{2}}(v+\sigmav)
v-={1\over{2}}(v-\sigmav)
Therefore, one gets a direct sum of vector spaces
V=V+ ⊕ V-
V+=\{v\inV|\sigmav=v\}
V-=\{v\inV|\sigmav=-v\}
Both sets
V+
V-
K:V+\toV-
K(t)=it
\dimRV+=\dimRV-=\dimCV
The first factor
V+
VR
\sigma
\sigma(VR)\subsetVR
V-
iVR
V=V+ ⊕ V-
V=VR ⊕ iVR
i.e. as the direct sum of the "real"
VR
iVR
VR
VC=VR ⊗ RC
VR
VR ⊗ RC=VR ⊕ iVR
It follows a natural linear isomorphism
VR ⊗ RC\toV
A real structure on a complex vector space V, that is an antilinear involution
\sigma:V\toV
\hat\sigma:V\to\barV
V
\barV
v\mapsto\hat\sigma(v):=\overline{\sigma(v)}
For an algebraic variety defined over a subfield of the real numbers,the real structure is the complex conjugation acting on the points of the variety in complex projective or affine space.Its fixed locus is the space of real points of the variety (which may be empty).
For a scheme defined over a subfield of the real numbers, complex conjugationis in a natural way a member of the Galois group of the algebraic closure of the base field.The real structure is the Galois action of this conjugation on the extension of thescheme over the algebraic closure of the base field.The real points are the points whose residue field is fixed (which may be empty).
In mathematics, a reality structure on a complex vector space V is a decomposition of V into two real subspaces, called the real and imaginary parts of V:
V=VR ⊕ iVR.
The standard reality structure on the vector space
Cn
Cn=Rn ⊕ iRn.
In the presence of a reality structure, every vector in V has a real part and an imaginary part, each of which is a vector in VR:
v=\operatorname{Re}\{v\}+i\operatorname{Im}\{v\}
\overlinev=\operatorname{Re}\{v\}-i\operatorname{Im}\{v\}
v\mapsto\overlinev
\overline{\overlinev}=v, \overline{v+w}=\overline{v}+\overline{w}, and \overline{\alphav}=\overline\alpha\overline{v}.
Conversely, given an antilinear involution
v\mapstoc(v)
\operatorname{Re}\{v\}= | 1 |
2 |
\left(v+c(v)\right),
VR=\left\{\operatorname{Re}\{v\}\midv\inV\right\}.
V=VR ⊕ iVR.
i