In mathematics, a real structure on a complex vector 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, giving the "canonical" real structure.

The conjugation map is antilinear.

Vector space

A real structure on a complex vector space V is an antilinear involution, its fixed locus, and the natural map

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 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. be an antilinear map, that is an antilinear involution of the complex space V.

Therefore, one gets a direct sum where:

1. .

are real vector spaces, is an isomorphism of real vector spaces, whence:

1. .

reads now as:

1. ,

parts of V. This construction strongly depends on the choice of an antilinear involution of the complex vector space V. The complexification admits a natural real structure:

1. .

between complex vector spaces with a given real structure.

A real structure on a complex vector space V, may be equivalently described in terms of the linear map to the complex conjugate vector space defined by

1. .

Algebraic variety

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).

Reality structure

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:

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 dimension n (real dimension 2n), then VR must have real dimension n.

The standard reality structure 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:

is an antilinear involution, i.e.

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 operator c. The eigenvalues of c are +1 and 1, with eigenspaces VRVR, respectively. Typically, the operator c itself, rather than the eigenspace decomposition it entails, is referred to as the reality structure on V.

See also

• Antilinear map

• Complex conjugate

• Complex conjugate vector space

• Complexification

• Linear complex structure

• Linear map

• Sesquilinear form

Notes

References

• Horn and Johnson, Matrix Analysis, Cambridge University Press, 1985. . (antilinear maps are discussed in section 4.6).

• Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988. . (antilinear maps are discussed in section 3.3).