Definitions

# Linear complex structure

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.

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.

## Definition and properties

A complex structure on a real vector space V is a real linear transformation

J : VV
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 the imaginary unit, 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 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 $\left(v_1, ldots, v_n\right)$ is a basis for the complex vector space V J then $\left(v_1, J v_1, ldots, v_n, J v_n\right)$ is a basis for the underlying real space V.

A real linear transformation A : VV 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

## Example

If V is any real vector space there is a canonical complex structure on the direct sum VV given by

$J\left(v,w\right) = \left(-w,v\right).,$
The block matrix form of J is
$J = begin\left\{bmatrix\right\}0 & -1 1 & 0end\left\{bmatrix\right\}$
where 1 is the identity map on V.

## 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.

## Relation to complexifications

Given any real vector space V we may define its complexification by
$V^\left\{mathbb C\right\}=Votimes_\left\{mathbb\left\{R\right\}\right\}mathbb\left\{C\right\}.$
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
$overline\left\{votimes z\right\} = votimesbar z$

If J is a complex structure on V, we may extend J by linearity to VC:

$J\left(votimes z\right) = J\left(v\right)otimes z.$

Since C is algebraically closed, J is guaranteed to have eigenvalues which satisfy λ2 = −1, namely λ = ±i. Thus we may write

$V^\left\{mathbb C\right\}= V^\left\{+\right\}oplus V^\left\{-\right\}$
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
$mathcal P^\left\{pm\right\} = \left\{1over 2\right\}\left(1mp iJ\right).$
So that
$V^\left\{pm\right\} = \left\{votimes 1 mp Jvotimes i: v in V\right\}.$

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:

$W^\left\{mathbb C\right\} cong Woplus overline\left\{W\right\}.$

## 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

$\left(V^*\right)^mathbb\left\{C\right\} = \left(V^*\right)^\left\{+\right\}oplus \left(V^*\right)^-$

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 = ST then the exterior powers of U can be decomposed as follows:

$Lambda^r U = bigoplus_\left\{p+q=r\right\}\left(Lambda^p S\right)otimes\left(Lambda^q T\right).$

A complex structure J on V therefore induces a decomposition

$Lambda^r,V^mathbb\left\{C\right\} = bigoplus_\left\{p+q=r\right\} Lambda^\left\{p,q\right\},V_J$
where
$Lambda^\left\{p,q\right\},V_J;stackrel\left\{mathrm\left\{def\right\}\right\}\left\{=\right\}, \left(Lambda^p,V^+\right)otimes\left(Lambda^p,V^-\right).$
All exterior powers are taken over the complex numbers. So if VJ is has complex dimension n (real dimension 2n) then

$dim_\left\{mathbb C\right\}Lambda^\left\{r\right\},V^\left\{mathbb C\right\} = \left\{2nchoose r\right\}qquad dim_\left\{mathbb C\right\}Lambda^\left\{p,q\right\},V_J = \left\{n choose p\right\}\left\{n choose q\right\}.$
The dimensions add up correctly as a consequence of Vandermonde's identity.

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.

See complex differential form and almost complex manifold for applications of these ideas.