In mathematics, a complex structure on a real vector space
allows one to define multiplication by complex scalars in a canonical fashion so as to regard
A complex structure on a real vector space
the structure of a complex vector space which we denote
Going in the other direction, if one starts with a complex vector space
, thought of as an associative algebra over the real numbers.
It is not hard to see that every even-dimensional vector space admits a complex structure.
, with matrix multiplication form complex numbers.
for the complex space, this set, together with these vectors multiplied by i, namely
There are two natural ways to order this basis, corresponding abstractly to whether one writes the tensor product 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
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.
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
If V is any real vector space there is a canonical complex structure on the direct sum V ⊕ V given by
is the identity map on V. This corresponds to the complex structure on the tensor product
Likewise, J preserves a nondegenerate, skew-symmetric form ω if and only if J is a symplectic transformation (that is, if
Given a symplectic form ω and a linear complex structure J on V, one may define an associated bilinear form gJ on V by
If in addition ω is tamed by J, then the associated form is positive definite.
Thus in this case V is an inner product space with respect to gJ.
If the symplectic form ω is preserved (but not necessarily tamed) by J, then gJ is the real part of the Hermitian form (by convention antilinear in the first argument)
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 VC: Since C is algebraically closed, J is guaranteed to have eigenvalues which satisfy λ2 = −1, namely λ = ±i.
The projection maps onto the V± eigenspaces are given by So that 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.
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 (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: A complex structure J on V therefore induces a decomposition where All exterior powers are taken over the complex numbers.
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.