Hermitian structure

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 (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 : 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(v,w) = (-w,v).,
The block matrix form of J is
J = begin{bmatrix}0 & -1 1 & 0end{bmatrix}
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^{mathbb C}=Votimes_{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{votimes z} = votimesbar z

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

J(votimes 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} = {1over 2}(1mp iJ).
So that
V^{pm} = {votimes 1 mp Jvotimes i: v in V}.

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^{mathbb C} cong Woplus overline{W}.

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 (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_{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 VJ is has complex dimension n (real dimension 2n) then

dim_{mathbb C}Lambda^{r},V^{mathbb C} = {2nchoose 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 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.

See also

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