V
V
-IdV
V
V
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 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.
V
J2
J
IdV
V
J
-1
i
V
x,y
\vec{v}
V
VJ
Going in the other direction, if one starts with a complex vector space
W
Jw=iw~~\forallw\inW
More formally, a linear complex structure on a real vector space is an algebra representation of the complex numbers
C
i2=-1
C
V
C
V
C → End(V)
i
i
i
End(V)
J
If
VJ
n
V
2n
V
J
e,f
Je=f
Jf=-e
V
(v1,...,vn)
VJ
(v1,Jv1,...,vn,Jvn)
V
A real linear transformation
A:V → V
VJ
A
J
U
V
VJ
J
U
The collection of
2 x 2
M(2,\Reals)
J=\begin{pmatrix}a&c\ b&-a\end{pmatrix},~~a2+bc=-1
M(2,\Reals)
I
xI+yJ
The fundamental example of a linear complex structure is the structure on R2n coming from the complex structure on Cn. That is, the complex n-dimensional space Cn is also a real 2n-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
i(λv)=(iλ)v=(λi)v=λ(iv)
Given a basis
\left\{e1,e2,...,en\right\}
\left\{ie1,ie2,...,ien\right\},
\Complexn=\Rn ⊗ \R\Complex
\Complexn=\Complex ⊗ \R\Rn.
If one orders the basis as
\left\{e1,ie1,e2,ie2,...,en,ien\right\},
\Complexm ⊕ \Complexn
\Complexm+n.
On the other hand, if one orders the basis as
\left\{e1,e2,...,en,ie1,ie2,...,ien\right\}
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 algebras and Lie groups, this corresponds to the inclusion of gl(n,C) in gl(2n,R) (Lie algebras – matrices, not necessarily invertible) and GL(n,C) in 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
[J,A]=0;
[J,-].
Note that the defining equations for these statements are the same, as
AJ=JA
AJ-JA=0,
[A,J]=0,
If V is any real vector space there is a canonical complex structure on the direct sum V ⊕ V given byThe block matrix form of J iswhere
IV
\Complex ⊗ \RV.
If is a bilinear form on then we say that preserves iffor 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 (that is, if ). For symplectic forms an interesting compatibility condition between and is thatholds 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 byBecause 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
Given any real vector space V we may define its complexification by extension of scalars:
VC=V ⊗ RC.
\overline{v ⊗ z}=v ⊗ \barz
If J is a complex structure on V, we may extend J by linearity to VC:
J(v ⊗ z)=J(v) ⊗ z.
Since C is algebraically closed, J is guaranteed to have eigenvalues which satisfy λ2 = −1, namely λ = ±i. Thus we may write
VC=V+ ⊕ V-
lP\pm={1\over2}(1\mpiJ).
V\pm=\{v ⊗ 1\mpJv ⊗ i:v\inV\}.
There is a natural complex linear isomorphism between VJ and V+, so these vector spaces can be considered the same, while V− may be regarded as the complex conjugate of VJ.
Note that if VJ has complex dimension n then both V+ and V− have complex dimension n while VC has complex dimension 2n.
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:
WC\congW ⊕ \overline{W}.
Let V be a real vector space with a complex structure J. The dual space 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
(V*)C=(V*)+ ⊕ (V*)-
into the ±i eigenspaces of J*. Under the natural identification of (V*)C with (VC)* 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, symmetric, and exterior algebras over VC 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:
ΛrU=oplusp+q=r(ΛpS) ⊗ (ΛqT).
A complex structure J on V therefore induces a decomposition
ΛrVC=oplusp+q=rΛp,qVJ
Λp,qVJ \stackrel{def
\dimCΛrVC={2n\chooser} \dimCΛp,qVJ={n\choosep}{n\chooseq}.
The space of (p,q)-forms Λp,q VJ* is the space of (complex) multilinear forms on VC which vanish on homogeneous elements unless p are from V+ and q are from V−. It is also possible to regard Λp,q VJ* as the space of real multilinear maps from VJ to C which are complex linear in p terms and conjugate-linear in q terms.
See complex differential form and almost complex manifold for applications of these ideas.