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 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 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 numbers , thought of as an associative algebra over the real numbers. 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 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 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 R2"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 formula_7 – 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 formula_8 for the complex space, this set, together with these vectors multiplied by "i," namely formula_9 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 formula_10 or instead as formula_11
If one orders the basis as formula_12 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 formula_14 is the same as that for formula_15
On the other hand, if one orders the basis as formula_16, 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 algebras 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):
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 formula_19 in other words, as the kernel of the map of bracketing with "J," formula_20
Note that the defining equations for these statements are the same, as formula_21 is the same as formula_22 which is the same as formula_23 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 formula_26 is the identity map on "V". This corresponds to the complex structure on the tensor product formula_27
Compatibility with other structures.
If is a bilinear form on then we say that preserves if
formula_28
for all . An equivalent characterization is that is skew-adjoint with respect to :
formula_29
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 formula_30). For symplectic forms an interesting compatibility condition between and is that
formula_31
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 formula_32 is tame).
Given a symplectic form and a linear complex structure on , one may define an associated bilinear form on by
formula_33
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) formula_34 defined by
formula_35
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 eigenvalues which satisfy λ2 = −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 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 "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, 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 and almost complex manifold for applications of these ideas.