In mathematics, a complex structure on a real vector space V is an automorphism of V which 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.
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 is 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 then 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.
If V is any real vector space there is a canonical complex structure on the direct sum V ⊕ V given by
If B is a bilinear form on V then we say that J preserves B if
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
Given a symplectic form ω and a linear complex structure J, one may define an associated symmetric bilinear form gJ on VJ
If J is a complex structure on V, we may extend J by linearity to VC:
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:
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 (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 decompositon U = S ⊕ T then the exterior powers of U can be decomposed as follows:
A complex structure J on V therefore induces a decomposition
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.