Linear complex structure

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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.

1 Definition and properties
2 Example
3 Compatibility with other structures
4 Relation to complexifications
5 Extension to related vector spaces
6 See also

[edit] Definition and properties

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

J : V → V

such that

J² = −id_V.

Here J² means J composed with itself and id_V 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 $(v_1, \ldots, v_n)$ is a basis for the complex vector space V _J then $(v_1, J v_1, \ldots, v_n, J v_n)$ 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

[edit] Example

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

$J(v,w) = (-w,v).\,$

The block matrix form of J is

$J = \begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}$

where 1 is the identity map on V.

[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 g_J on V_J

g_J(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 V_J is an inner product space.

[edit] Relation to complexifications

Given any real vector space V we may define its complexification by

$V^{\mathbb C}=V\otimes_{\mathbb{R}}\mathbb{C}.$

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

$\overline{v\otimes z} = v\otimes\bar z$

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

$J(v\otimes z) = J(v)\otimes z.$

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

$V^{\mathbb C}= V^{+}\oplus V^{-}$

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

$\mathcal P^{\pm} = {1\over 2}(1\mp iJ).$

So that

$V^{\pm} = \{v\otimes 1 \mp Jv\otimes i: v \in V\}.$

There is a natural complex linear isomorphism between V_J and V⁺, so these vector spaces can be considered the same, while V⁻ may be regarded as the complex conjugate of V_J.

Note that if V_J has complex dimension n then both V⁺ and V⁻ have complex dimension n while V^C 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:

$W^{\mathbb C} \cong W\oplus \overline{W}.$

[edit] 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

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

into the ±i eigenspaces of J*. Under the natural identification of (V*)^C with (V^C)* 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 V^C 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:

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

A complex structure J on V therefore induces a decomposition

$\Lambda^r\,V^\mathbb{C} = \bigoplus_{p+q=r} \Lambda^{p,q}\,V_J$

where

$\Lambda^{p,q}\,V_J\;\stackrel{\mathrm{def}}{=}\, (\Lambda^p\,V^+)\otimes(\Lambda^p\,V^-).$

All exterior powers are taken over the complex numbers. So if V_J is has complex dimension n (real dimension 2n) then

$\dim_{\mathbb C}\Lambda^{r}\,V^{\mathbb C} = {2n\choose r}\qquad \dim_{\mathbb C}\Lambda^{p,q}\,V_J = {n \choose p}{n \choose q}.$

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

The space of (p,q)-forms Λ^p,q V_J* is the space of (complex) multilinear forms on V^C which vanish on homogeneous elements unless p are from V⁺ and q are from V⁻. It is also possible to regard Λ^p,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.