Linear Complex Structure - Extension To Related Vector Spaces

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 (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. So if V_J is has complex dimension n (real dimension 2n) then

The dimensions add up correctly as a consequence of Vandermonde's identity.

The space of (p,q)-forms Λp,q V_J* 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 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.