Linear complex structure
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In mathematics, a complex structure on a real vector space V is a real linear transformation
- J : V → V
such that
- J2 = −idV.
Here J2 means J composed with itself and idV is the identity map on V. That is, the effect of applying J twice is the same as multiplication by −1. This is reminiscent of multiplication by i. A complex structure allows one to endow V with the structure of a complex vector space. Complex scalar multiplication can be defined by
- (x + i y)v = xv + yJ(v)
for all real numbers x,y and all vectors v in V. One can check that this does, in fact, give V the structure of a complex vector space which we denote V J.
Going in the other direction, if one starts with a complex vector space W then one can define a complex structure on the underlying real space by defining Jw = i w for all w in W.
If V J has complex dimension n then V must have real dimension 2n. That is, V admits a complex structure only if it even-dimensional. It is not hard to see that every even-dimensional vector space admits a complex structure. (One can define J on pairs e,f of basis vectors by Je = f and Jf = −e and the extend by linearity to all of V). If 
is a basis for the complex vector space V J then 
is a basis for the underlying real space V.
A real linear transformation A : V → V is a complex linear transformation of the corresponding complex space V J if and only if A commutes with J, i.e.
- AJ = JA
Likewise, a real subspace U of V is a complex subspace of V J if and only if J preserves U, i.e.
- JU = U
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[edit] Relation to complexifications
If J is a complex structure on V, we may extend J by linearity to the complexification of V,
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−, where V+ and V− are the eigenspaces of +i and −i, respectively. Complex conjugation provides a conjugate-linear isomorphism over C between V+ and V−, and thus they have the same complex dimension. Thus if n is the complex dimension of V+, then 2n is the complex dimension of VC, and so 2n is also the real dimension of V. Here V+ is the subspace of the complexification of V that we defined above, while V− is the complex space on which J acts as multiplication by −i.
Note that there is a complex linear isomorphism between V J and V+, so these vector spaces can be considered the same.
[edit] Compatibility with other structures
If B is a bilinear form on V then we say that J preserves B if
- B(Ju, Jv) = B(u, v)
for all u,v in V. An equivalent characterization is that J is skew-adjoint with respect to B:
- B(Ju, v) = −B(u, Jv)
If g is an inner product on V then J preserves g if and only if J is an orthogonal transformation. Likewise, J preserves a nondegenerate, skew-symmetric form ω if and only if J is a symplectic transformation (that is, if ω(Ju,Jv) = ω(u,v)). For symplectic forms ω there is usually an added restriction for compatibility between J and ω, namely
- ω(u, Ju) > 0
for all u in V. If this condition is satisfied then J is said to tame ω.
Given a symplectic form ω and a linear complex structure J, one may define an associated symmetric bilinear form gJ on VJ
- gJ(u,v) = ω(u,Jv).
Because a symplectic form is nondegenerate, so is the associated bilinear form. Moreover, the associated form is preserved by J if and only if the symplectic form and if ω is tamed by J then the associated form is positive definite. Thus in this case the associated form is an Hermitian form and VJ is an inner product space.
[edit] Extension to related vector spaces
As discussed above, a linear complex structure on a real vector space induces the decomposition
on the complexification of V. This decomposition can be extended to several vector spaces built from V. For example, the dual space of V can also be decomposed into functionals of type (1,0) and (0,1). Functionals of type (1,0) are those which vanish on V-. The tensor algebra, symmetric algebra, and exterior algebra over V also admit decompositions. The exterior algebra is perhaps the most important application of this decomposition. The homogeneous spaces of k-vectors decompose
and the entire exterior algebra becomes
where n is the real dimension of the real vector space (and so also the complex dimension of its complexification), and Λ(p,q) is the exterior power of p copies of V(1,0) and q copies of V(0,1). See almost complex manifold for an application of this construction. Similar expansions hold for the tensor algebra and the symmetric algebra.
