In
mathematics, a real structure on a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
is a way to decompose the complex vector space in the
direct sum of two
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
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
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
, with
, giving the "canonical" real structure on
, that is
.
The conjugation map is
antilinear
In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if
\begin
f(x + y) &= f(x) + f(y) && \qquad \text \\
f(s x) &= \overline f(x) && \qquad \text \\
\end
hold for all vectors x, y ...
:
and
.
Vector space
A real structure on a
complex vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V'' is an
antilinear
In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if
\begin
f(x + y) &= f(x) + f(y) && \qquad \text \\
f(s x) &= \overline f(x) && \qquad \text \\
\end
hold for all vectors x, y ...
involution
Involution may refer to:
* Involute, a construction in the differential geometry of curves
* '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
. A real structure defines a real subspace
, its fixed locus, and the natural map
:
is an isomorphism. Conversely any vector space that is the
complexification
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include ...
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 theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
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. Let
be an
antilinear map
In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if
\begin
f(x + y) &= f(x) + f(y) && \qquad \text \\
f(s x) &= \overline f(x) && \qquad \text \\
\end
hold for all vectors x, y ...
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 space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s. 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
antilinear
In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if
\begin
f(x + y) &= f(x) + f(y) && \qquad \text \\
f(s x) &= \overline f(x) && \qquad \text \\
\end
hold for all vectors x, y ...
involution
Involution may refer to:
* Involute, a construction in the differential geometry of curves
* '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
of the complex vector space ''V''. The
complexification
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include ...
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
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
from the vector space
to the
complex conjugate vector space
In mathematics, the complex conjugate of a complex vector space V\, is a complex vector space \overline V, which has the same elements and additive group structure as V, but whose scalar multiplication involves conjugation of the scalars. In other ...
defined by
:
.
[Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Springer-Verlag, 1988, p. 29.]
Algebraic variety
For an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
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
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
of the
algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V'' is a decomposition of ''V'' into two real subspaces, called the
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
and
imaginary part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s of ''V'':
:
Here ''V''
R is a real subspace of ''V'', i.e. a subspace of ''V'' considered as a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. If ''V'' has
complex dimension In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local neighborhoods of points (or of non-singular points in the case of a variety) are modeled on a ...
''n'' (real dimension 2''n''), then ''V''
R 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 ''V''
R:
:
In this case, the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of a vector ''v'' is defined as follows:
:
This map
is an
antilinear
In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if
\begin
f(x + y) &= f(x) + f(y) && \qquad \text \\
f(s x) &= \overline f(x) && \qquad \text \\
\end
hold for all vectors x, y ...
involution
Involution may refer to:
* Involute, a construction in the differential geometry of curves
* '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
, i.e.
:
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
eigenspace
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s of the real
linear operator ''c''. The eigenvalues of ''c'' are +1 and −1, with eigenspaces ''V''
R and
''V''
R, 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
In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if
\begin
f(x + y) &= f(x) + f(y) && \qquad \text \\
f(s x) &= \overline f(x) && \qquad \text \\
\end
hold for all vectors x, y ...
*
Canonical complex conjugation map
*
Complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
*
Complex conjugate vector space
In mathematics, the complex conjugate of a complex vector space V\, is a complex vector space \overline V, which has the same elements and additive group structure as V, but whose scalar multiplication involves conjugation of the scalars. In other ...
*
Complexification
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include ...
*
Linear complex structure
In mathematics, a complex structure on a real vector space ''V'' is an automorphism of ''V'' that squares to the minus identity, −''I''. Such a structure on ''V'' allows one to define multiplication by complex scalars in a canonical fashion so ...
*
Linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
*
Sesquilinear form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows o ...
*
Spinor calculus
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).
* {{Citation , last1=Penrose , first1=Roger , author1-link=Roger Penrose , last2=Rindler , first2=Wolfgang , author2-link=Wolfgang Rindler , title=Spinors and space-time. Vol. 2 , publisher=
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer.
Cambridge University Pre ...
, series=Cambridge Monographs on Mathematical Physics , isbn=978-0-521-25267-6 , mr=838301 , year=1986
Structures on manifolds