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 as to regard ''V'' as a complex vector space.
Every complex vector space can be equipped with a compatible complex structure, however, there is in general no canonical such structure. Complex structures have applications in
representation theory as well as in
complex geometry where they play an essential role in the definition of
almost complex manifolds, by contrast to
complex manifolds. 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 ''V'' is a real
linear transformation
:
such that
:
Here means
composed
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
with itself and is the
identity map on . That is, the effect of applying twice is the same as multiplication by . This is reminiscent of multiplication by the
imaginary unit, . 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 for all .
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 over the
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 coor ...
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" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
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
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 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 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 iden ...
s and
Lie groups, 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):
:gl(''n'',C) < gl(''2n'',R) and GL(''n'',C) < GL(''2n'',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 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 ''V'' ⊕ ''V'' given by
:
The
block matrix 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 lin ...
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 on then preserves if and only if is an
orthogonal transformation. Likewise, preserves a
nondegenerate,
skew-symmetric form if and only if is a
symplectic transformation In mathematics, a symplectic matrix is a 2n\times 2n matrix M with real entries that satisfies the condition
where M^\text denotes the transpose of M and \Omega is a fixed 2n\times 2n nonsingular, skew-symmetric matrix. This definition can be ext ...
(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 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 with respect to .
If the symplectic form is preserved (but not necessarily tamed) by , then is the
real part of the
Hermitian form (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 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 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 denot ...
s which satisfy λ
2 = −1, namely λ = ±''i''. Thus we may write
:
where ''V''
+ and ''V''
− are the
eigenspaces 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, then) the complex conjugate of a + bi is equal to 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 con ...
''V''* has a natural complex structure ''J''* given by the dual (or
transpose) 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 related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...
,
symmetric, and
exterior algebras 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 forms 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 maps from ''V''
''J'' to C which are complex linear in ''p'' terms and
conjugate-linear in ''q'' terms.
See
complex differential form
In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients.
Complex forms have broad applications in differential geometry. On complex manifolds, ...
and
almost complex manifold for applications of these ideas.
See also
*
Almost complex manifold
*
Complex manifold
*
Complex differential form
In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients.
Complex forms have broad applications in differential geometry. On complex manifolds, ...
*
Complex conjugate vector space
*
Hermitian structure
*
Real structure
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 compl ...
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