In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a
complex structure on a
real vector space
Real may refer to:
Currencies
* Argentine real
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Nature and science
* Reality, the state of things as they exist, ...
is an
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...
of
that squares to the minus
identity,
. Such a structure on
allows one to define multiplication by
complex scalars in a canonical fashion so as to regard
as a
complex vector space.
Every complex vector space can be equipped with a compatible complex structure in a canonical way; however, there is in general no canonical complex structure. Complex structures have applications in
representation theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
as well as in
complex geometry
In mathematics, complex geometry is the study of geometry, geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of space (mathematics), spaces su ...
where they play an essential role in the definition of
almost complex manifold
In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not comple ...
s, by contrast to
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
s. The term "complex structure" often refers to this structure on manifolds; when it refers instead to a structure on vector spaces, it may be called a linear complex structure.
Definition and properties
A complex structure on a
real vector space
Real may refer to:
Currencies
* Argentine real
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Nature and science
* Reality, the state of things as they exist, ...
is a real
linear transformation
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 ...
such that
Here
means
composed with itself and
is the
identity map
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
on
. That is, the effect of applying
twice is the same as multiplication by
. This is reminiscent of multiplication by the
imaginary unit
The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
,
. A complex structure allows one to endow
with the structure of a
complex vector space. Complex scalar multiplication can be defined by
for all real numbers
and all vectors
in . One can check that this does, in fact, give
the structure of a complex vector space which we denote
.
Going in the other direction, if one starts with a complex vector space
then one can define a complex structure on the underlying real space by defining
.
More formally, a linear complex structure on a real vector space is an
algebra representation of the
complex number
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 for ...
s
, thought of as an
associative algebra
In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...
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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s. This algebra is realized concretely as
which corresponds to
. Then a representation of
is a real vector space
, together with an action of
on
(a map
). Concretely, this is just an action of
, as this generates the algebra, and the operator representing
(the image of
in
) is exactly
.
If
has complex
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
, then
must have real dimension
. That is, a finite-dimensional space
admits a complex structure only if it is even-dimensional. It is not hard to see that every even-dimensional vector space admits a complex structure. One can define
on pairs
of
basis vectors by
and
and then
extend by linearity to all of
. If
is a basis for the complex vector space
then
is a basis for the underlying real space
.
A real linear transformation
is a ''complex'' linear transformation of the corresponding complex space
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
commutes with
, i.e. if and only if
Likewise, a real
subspace of
is a complex subspace of
if and only if
preserves
, i.e. if and only if
Examples
Elementary example
The collection of
real matrices
over the real field is 4-dimensional. Any matrix
:
has square equal to the negative of the identity matrix. A complex structure may be formed in
: with identity matrix
, elements
, with
matrix multiplication
In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
form complex numbers.
Complex ''n''-dimensional space C''n''
The fundamental example of a linear complex structure is the structure on R
2''n'' coming from the complex structure on C
''n''. That is, the complex ''n''-dimensional space C
''n'' is also a real 2''n''-dimensional space – using the same vector addition and real scalar multiplication – while multiplication by the complex number ''i'' is not only a ''complex'' linear transform of the space, thought of as a complex vector space, but also a ''real'' linear transform of the space, thought of as a real vector space. Concretely, this is because scalar multiplication by ''i'' commutes with scalar multiplication by real numbers
– and distributes across vector addition. As a complex ''n''×''n'' matrix, this is simply the
scalar matrix
In linear algebra, a diagonal matrix is a matrix (mathematics), matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An exampl ...
with ''i'' on the diagonal. The corresponding real 2''n''×2''n'' matrix is denoted ''J''.
Given a basis
for the complex space, this set, together with these vectors multiplied by ''i,'' namely
form a basis for the real space. There are two natural ways to order this basis, corresponding abstractly to whether one writes the tensor product as
or instead as
If one orders the basis as
then the matrix for ''J'' takes the
block diagonal form (subscripts added to indicate dimension):
This ordering has the advantage that it respects direct sums of complex vector spaces, meaning here that the basis for
is the same as that for
On the other hand, if one orders the basis as
, then the matrix for ''J'' is block-antidiagonal:
This ordering is more natural if one thinks of the complex space as a
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
of real spaces, as discussed below.
The data of the real vector space and the ''J'' matrix is exactly the same as the data of the complex vector space, as the ''J'' matrix allows one to define complex multiplication. At the level of
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
s and
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
s, this corresponds to the inclusion of gl(''n'',C) in gl(2''n'',R) (Lie algebras – matrices, not necessarily invertible) and
GL(''n'',C) in GL(2''n'',R):
The inclusion corresponds to forgetting the complex structure (and keeping only the real), while the subgroup GL(''n'',C) can be characterized (given in equations) as the matrices that ''commute'' with ''J:''
The corresponding statement about Lie algebras is that the subalgebra gl(''n'',C) of complex matrices are those whose
Lie bracket
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...
with ''J'' vanishes, meaning
in other words, as the kernel of the map of bracketing with ''J,''
Note that the defining equations for these statements are the same, as
is the same as
which is the same as
though the meaning of the Lie bracket vanishing is less immediate geometrically than the meaning of commuting.
Direct sum
If ''V'' is any real vector space there is a canonical complex structure on the
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
''V'' ⊕ ''V'' given by
The
block matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices.
Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix w ...
form of ''J'' is
where
is the identity map on ''V''. This corresponds to the complex structure on the tensor product
Compatibility with other structures
If is a
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
on then we say that preserves if
for all . An equivalent characterization is that is
skew-adjoint with respect to :
If is an
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
on then preserves if and only if is an
orthogonal transformation
In linear algebra, an orthogonal transformation is a linear transformation ''T'' : ''V'' → ''V'' on a real inner product space ''V'', that preserves the inner product. That is, for each pair of elements of ''V'', we hav ...
. Likewise, preserves a
nondegenerate,
skew-symmetric form if and only if is a
symplectic transformation (that is, if
). For symplectic forms an interesting compatibility condition between and is that
holds for all non-zero in . If this condition is satisfied, then we say that tames (synonymously: that is tame with respect to ; that is tame with respect to ; or that the pair
is tame).
Given a symplectic form and a linear complex structure on , one may define an associated bilinear form on by
Because a
symplectic form
In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers \mathbb) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping \omega : V \times V \to F that is
; Bilinear: ...
is nondegenerate, so is the associated bilinear form. The associated form is preserved by if and only if the symplectic form is. Moreover, if the symplectic form is preserved by , then the associated form is symmetric. If in addition is tamed by , then the associated form is
positive definite. Thus in this case is an
inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
with respect to .
If the symplectic form is preserved (but not necessarily tamed) by , then is the
real 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 ...
of the
Hermitian 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 map, linear in each of its arguments, but a sesquilinear f ...
(by convention antilinear in the first argument)
defined by
Relation to complexifications
Given any real vector space ''V'' we may define its
complexification by
extension of scalars:
:
This is a complex vector space whose complex dimension is equal to the real dimension of ''V''. It has a canonical
complex conjugation
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 numbers, then the complex conjugate of a + bi is a - ...
defined by
:
If ''J'' is a complex structure on ''V'', we may extend ''J'' by linearity to ''V''
C:
:
Since C is
algebraically closed, ''J'' is guaranteed to have
eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s which satisfy λ
2 = −1, namely λ = ±''i''. Thus we may write
:
where ''V''
+ and ''V''
− are the
eigenspace
In linear algebra, an eigenvector ( ) or characteristic vector is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an e ...
s of +''i'' and −''i'', respectively. Complex conjugation interchanges ''V''
+ and ''V''
−. The projection maps onto the ''V''
± eigenspaces are given by
:
So that
:
There is a natural complex linear isomorphism between ''V''
''J'' and ''V''
+, so these vector spaces can be considered the same, while ''V''
− may be regarded as 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 numbers, then the complex conjugate of a + bi is a - ...
of ''V''
''J''.
Note that if ''V''
''J'' has complex dimension ''n'' then both ''V''
+ and ''V''
− have complex dimension ''n'' while ''V''
C has complex dimension 2''n''.
Abstractly, if one starts with a complex vector space ''W'' and takes the complexification of the underlying real space, one obtains a space isomorphic to the direct sum of ''W'' and its conjugate:
:
Extension to related vector spaces
Let ''V'' be a real vector space with a complex structure ''J''. The
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by cons ...
''V''* has a natural complex structure ''J''* given by the dual (or
transpose
In linear algebra, the transpose of a Matrix (mathematics), matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other ...
) of ''J''. The complexification of the dual space (''V''*)
C therefore has a natural decomposition
:
into the ±''i'' eigenspaces of ''J''*. Under the natural identification of (''V''*)
C with (''V''
C)* one can characterize (''V''*)
+ as those complex linear functionals which vanish on ''V''
−. Likewise (''V''*)
− consists of those complex linear functionals which vanish on ''V''
+.
The (complex)
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
,
symmetric
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, and
exterior algebra
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
s over ''V''
C also admit decompositions. The exterior algebra is perhaps the most important application of this decomposition. In general, if a vector space ''U'' admits a decomposition ''U'' = ''S'' ⊕ ''T'', then the exterior powers of ''U'' can be decomposed as follows:
:
A complex structure ''J'' on ''V'' therefore induces a decomposition
:
where
:
All exterior powers are taken over the complex numbers. So if ''V''
''J'' has complex dimension ''n'' (real dimension 2''n'') then
:
The dimensions add up correctly as a consequence of
Vandermonde's identity.
The space of (''p'',''q'')-forms Λ
''p'',''q'' ''V''
''J''* is the space of (complex)
multilinear form
In abstract algebra and multilinear algebra, a multilinear form on a vector space V over a field K is a map
:f\colon V^k \to K
that is separately K- linear in each of its k arguments. More generally, one can define multilinear forms on a mo ...
s on ''V''
C which vanish on homogeneous elements unless ''p'' are from ''V''
+ and ''q'' are from ''V''
−. It is also possible to regard Λ
''p'',''q'' ''V''
''J''* as the space of real
multilinear map
Multilinear may refer to:
* Multilinear form, a type of mathematical function from a vector space to the underlying field
* Multilinear map, a type of mathematical function between vector spaces
* Multilinear algebra, a field of mathematics ...
s from ''V''
''J'' to C which are complex linear in ''p'' terms and
conjugate-linear in ''q'' terms.
See
complex differential form and
almost complex manifold
In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not comple ...
for applications of these ideas.
See also
*
Almost complex manifold
In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not comple ...
*
Complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
*
Complex differential form
*
Complex conjugate vector space
*
Hermitian structure
*
Real structure
References
* Kobayashi S. and Nomizu K.,
Foundations of Differential Geometry, John Wiley & Sons, 1969. . (complex structures are discussed in Volume II, Chapter IX, section 1).
* Budinich, P. and Trautman, A. ''The Spinorial Chessboard'', Springer-Verlag, 1988. . (complex structures are discussed in section 3.1).
* Goldberg S.I., ''Curvature and Homology'', Dover Publications, 1982. {{isbn, 0-486-64314-X. (complex structures and almost complex manifolds are discussed in section 5.2).
Structures on manifolds