In
mathematics, a real structure on a
complexvector space is a way to decompose the complex vector space in the
direct sum of two
real vector spaces. The prototype of such a structure is the field of complex numbers itself, considered as a complex vector space over itself and with the conjugation
map, with , giving the "canonical" real structure on , that is .
is an isomorphism. Conversely any vector space that is the
complexification
of 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 and then the vectors and are
linearly independent in the realification of V. Hence:
Naturally, one would wish to represent V as the direct sum of two real vector spaces, the "real and imaginary parts of V". There is no canonical way of doing this: such a splitting is an additional real structure in V. It may be introduced as follows.[1] Let be an
antilinear map such that , that is an antilinear involution of the complex space V.
Any vector can be written ,
where and .
Therefore, one gets a
direct sum of vector spaces where:
and .
Both sets and are real
vector spaces. The linear map , where , is an isomorphism of real vector spaces, whence:
.
The first factor is also denoted by and is left invariant by , that is . The second factor is
usually denoted by . The direct sum reads now as:
,
i.e. as the direct sum of the "real" and "imaginary" parts of V. This construction strongly depends on the choice of an
antilinearinvolution of the complex vector space V. The
complexification of the real vector space , i.e.,
admits
a natural real structure and hence is canonically isomorphic to the direct sum of two copies of :
.
It follows a natural linear isomorphism between complex vector spaces with a given real structure.
A real structure on a complex vector space V, that is an antilinear involution , may be equivalently described in terms of the
linear map from the vector space to the
complex conjugate vector space defined by
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).
Scheme
For a scheme defined over a subfield of the real numbers, complex conjugation
is 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 the
scheme over the algebraic closure of the base field.
The real points are the points whose residue field is fixed (which may be empty).
Here VR is a real subspace of V, i.e. a subspace of V considered as a
vector space over the
real numbers. If V has
complex dimensionn (real dimension 2n), then VR must have real dimension n.
The standard reality structure on the vector space is the decomposition
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:
In this case, the
complex conjugate of a vector v is defined as follows:
Conversely, given an antilinear involution on a complex vector space V, it is possible to define a reality structure on V as follows. Let
and define
Then
This is actually the decomposition of V as the
eigenspaces of the real
linear operatorc. The eigenvalues of c are +1 and −1, with eigenspaces VR and VR, respectively. Typically, the operator c itself, rather than the eigenspace decomposition it entails, is referred to as the reality structure on V.