Linear complex structure

<h2 id="definition-and-properties">Definition and properties</h2>
A complex structure on a <a href="/facts/Real_vector_space/M1pxMLx2">real vector space</a> 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is a real <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a>

J
        :
        V
        →
        V
      
    
    {\displaystyle J:V\to V}

such that

J
          
            2
          
        
        =
        −
        
          
            id
          
          
            V
          
        
        .
      
    
    {\displaystyle J^{2}=-{\text{id}}_{V}.}

Here 
 
 
 
 
 J
 
 2
 
 
 
 
 {\displaystyle J^{2}}
 
 means 
 
 
 
 J
 
 
 {\displaystyle J}
 
 <a href="/facts/Function_composition/r5Pvnmth">composed</a> with itself and 
 
 
 
 
 
 id
 
 
 V
 
 
 
 
 {\displaystyle {\text{id}}_{V}}
 
 is the <a href="/facts/Identity_function/9AtsP0LC">identity map</a> on 
 
 
 
 V
 
 
 {\displaystyle V}
 
. That is, the effect of applying 
 
 
 
 J
 
 
 {\displaystyle J}
 
 twice is the same as multiplication by 
 
 
 
 −
 1
 
 
 {\displaystyle -1}
 
. This is reminiscent of multiplication by the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a>, 
 
 
 
 i
 
 
 {\displaystyle i}
 
. A complex structure allows one to endow 
 
 
 
 V
 
 
 {\displaystyle V}
 
 with the structure of a <a href="/facts/Complex_vector_space/M1pxMLx2">complex vector space</a>. Complex scalar multiplication can be defined by

(
        x
        +
        i
        y
        )
        
          
            
              v
              →
            
          
        
        =
        x
        
          
            
              v
              →
            
          
        
        +
        y
        J
        (
        
          
            
              v
              →
            
          
        
        )
      
    
    {\displaystyle (x+iy){\vec {v}}=x{\vec {v}}+yJ({\vec {v}})}

for all real numbers 
 
 
 
 x
 ,
 y
 
 
 {\displaystyle x,y}
 
 and all vectors 
 
 
 
 
 
 
 v
 →
 
 
 
 
 
 {\displaystyle {\vec {v}}}
 
 in V. One can check that this does, in fact, give 
 
 
 
 V
 
 
 {\displaystyle V}
 
 the structure of a complex vector space which we denote 
 
 
 
 
 V
 
 J
 
 
 
 
 {\displaystyle V_{J}}
 
.
Going in the other direction, if one starts with a complex vector space 
 
 
 
 W
 
 
 {\displaystyle W}
 
 then one can define a complex structure on the underlying real space by defining 
 
 
 
 J
 w
 =
 i
 w
  
  
 ∀
 w
 ∈
 W
 
 
 {\displaystyle Jw=iw~~\forall w\in W}
 
.
More formally, a linear complex structure on a real vector space is an <a href="/facts/Algebra_representation/1MaaEjqH">algebra representation</a> of the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
, thought of as an <a href="/facts/Associative_algebra/DRfoF6KV">associative algebra</a> over the <a href="/facts/Real_number/R02gw5Pb">real numbers</a>. This algebra is realized concretely as

C
        
        =
        
          R
        
        [
        x
        ]
        
          /
        
        (
        
          x
          
            2
          
        
        +
        1
        )
        ,
      
    
    {\displaystyle \mathbb {C} =\mathbb {R} [x]/(x^{2}+1),}

which corresponds to 
 
 
 
 
 i
 
 2
 
 
 =
 −
 1
 
 
 {\displaystyle i^{2}=-1}
 
. Then a representation of 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 is a real vector space 
 
 
 
 V
 
 
 {\displaystyle V}
 
, together with an action of 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 on 
 
 
 
 V
 
 
 {\displaystyle V}
 
 (a map 
 
 
 
 
 C
 
 →
 
 End
 
 (
 V
 )
 
 
 {\displaystyle \mathbb {C} \rightarrow {\text{End}}(V)}
 
). Concretely, this is just an action of 
 
 
 
 i
 
 
 {\displaystyle i}
 
, as this generates the algebra, and the operator representing 
 
 
 
 i
 
 
 {\displaystyle i}
 
 (the image of 
 
 
 
 i
 
 
 {\displaystyle i}
 
 in 
 
 
 
 
 End
 
 (
 V
 )
 
 
 {\displaystyle {\text{End}}(V)}
 
) is exactly 
 
 
 
 J
 
 
 {\displaystyle J}
 
.
If 
 
 
 
 
 V
 
 J
 
 
 
 
 {\displaystyle V_{J}}
 
 has complex <a href="/facts/Dimension_(linear_algebra)/z8m02A3W">dimension</a> 
 
 
 
 n
 
 
 {\displaystyle n}
 
, then 
 
 
 
 V
 
 
 {\displaystyle V}
 
 must have real dimension 
 
 
 
 2
 n
 
 
 {\displaystyle 2n}
 
. That is, a finite-dimensional space 
 
 
 
 V
 
 
 {\displaystyle 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
 
 
 {\displaystyle J}
 
 on pairs 
 
 
 
 e
 ,
 f
 
 
 {\displaystyle e,f}
 
 of <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a> vectors by 
 
 
 
 J
 e
 =
 f
 
 
 {\displaystyle Je=f}
 
 and 
 
 
 
 J
 f
 =
 −
 e
 
 
 {\displaystyle Jf=-e}
 
 and then <a href="/facts/Extend_by_linearity/Q1PBKNVV">extend by linearity</a> to all of 
 
 
 
 V
 
 
 {\displaystyle V}
 
. If 
 
 
 
 (
 
 v
 
 1
 
 
 ,
 …
 ,
 
 v
 
 n
 
 
 )
 
 
 {\displaystyle (v_{1},\dots ,v_{n})}
 
 is a basis for the complex vector space 
 
 
 
 
 V
 
 J
 
 
 
 
 {\displaystyle V_{J}}
 
 then 
 
 
 
 (
 
 v
 
 1
 
 
 ,
 J
 
 v
 
 1
 
 
 ,
 …
 ,
 
 v
 
 n
 
 
 ,
 J
 
 v
 
 n
 
 
 )
 
 
 {\displaystyle (v_{1},Jv_{1},\dots ,v_{n},Jv_{n})}
 
 is a basis for the underlying real space 
 
 
 
 V
 
 
 {\displaystyle V}
 
.
A real linear transformation 
 
 
 
 A
 :
 V
 →
 V
 
 
 {\displaystyle A:V\rightarrow V}
 
 is a complex linear transformation of the corresponding complex space 
 
 
 
 
 V
 
 J
 
 
 
 
 {\displaystyle V_{J}}
 
 <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> 
 
 
 
 A
 
 
 {\displaystyle A}
 
 commutes with 
 
 
 
 J
 
 
 {\displaystyle J}
 
, i.e. if and only if

A
        J
        =
        J
        A
        .
      
    
    {\displaystyle AJ=JA.}

Likewise, a real <a href="/facts/Linear_subspace/2wtKTgHL">subspace</a> 
 
 
 
 U
 
 
 {\displaystyle U}
 
 of 
 
 
 
 V
 
 
 {\displaystyle V}
 
 is a complex subspace of 
 
 
 
 
 V
 
 J
 
 
 
 
 {\displaystyle V_{J}}
 
 if and only if 
 
 
 
 J
 
 
 {\displaystyle J}
 
 preserves 
 
 
 
 U
 
 
 {\displaystyle U}
 
, i.e. if and only if

J
        U
        =
        U
        .
      
    
    {\displaystyle JU=U.}

<h2 id="examples">Examples</h2>
<h3>Elementary example</h3>
The collection of 
 
 
 
 2
 ×
 2
 
 
 {\displaystyle 2\times 2}
 
 real matrices 
 
 
 
 
 M
 
 (
 2
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathbb {M} (2,\mathbb {R} )}
 
 over the real field is 4-dimensional. Any matrix

J
        =
        
          
            (
            
              
                
                  a
                
                
                  c
                
              
              
                
                  b
                
                
                  −
                  a
                
              
            
            )
          
        
        ,
         
         
        
          a
          
            2
          
        
        +
        b
        c
        =
        −
        1
      
    
    {\displaystyle J={\begin{pmatrix}a&c\\b&-a\end{pmatrix}},~~a^{2}+bc=-1}

has square equal to the negative of the identity matrix. A complex structure may be formed in 
 
 
 
 
 M
 
 (
 2
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathbb {M} (2,\mathbb {R} )}
 
: with identity matrix 
 
 
 
 I
 
 
 {\displaystyle I}
 
, elements 
 
 
 
 x
 I
 +
 y
 J
 
 
 {\displaystyle xI+yJ}
 
, with <a href="/facts/Matrix_multiplication/lYDB6Ro2">matrix multiplication</a> form complex numbers.

<h3>Complex n-dimensional space Cn</h3>
The fundamental example of a linear complex structure is the structure on R2n coming from the complex structure on Cn. That is, the complex n-dimensional space Cn is also a real 2n-dimensional space – using the same vector addition and real scalar multiplication – while multiplication by the complex number i is not only a complex linear transform of the space, thought of as a complex vector space, but also a real linear transform of the space, thought of as a real vector space. Concretely, this is because scalar multiplication by i commutes with scalar multiplication by real numbers 
 
 
 
 i
 (
 λ
 v
 )
 =
 (
 i
 λ
 )
 v
 =
 (
 λ
 i
 )
 v
 =
 λ
 (
 i
 v
 )
 
 
 {\displaystyle i(\lambda v)=(i\lambda )v=(\lambda i)v=\lambda (iv)}
 
 – and distributes across vector addition. As a complex n×n matrix, this is simply the <a href="/facts/Scalar_matrix/tjIAHrE6">scalar matrix</a> with i on the diagonal. The corresponding real 2n×2n matrix is denoted J.
Given a basis 
 
 
 
 
 {
 
 
 e
 
 1
 
 
 ,
 
 e
 
 2
 
 
 ,
 …
 ,
 
 e
 
 n
 
 
 
 }
 
 
 
 {\displaystyle \left\{e_{1},e_{2},\dots ,e_{n}\right\}}
 
 for the complex space, this set, together with these vectors multiplied by i, namely 
 
 
 
 
 {
 
 i
 
 e
 
 1
 
 
 ,
 i
 
 e
 
 2
 
 
 ,
 …
 ,
 i
 
 e
 
 n
 
 
 
 }
 
 ,
 
 
 {\displaystyle \left\{ie_{1},ie_{2},\dots ,ie_{n}\right\},}
 
 form a basis for the real space. There are two natural ways to order this basis, corresponding abstractly to whether one writes the tensor product as 
 
 
 
 
 
 C
 
 
 n
 
 
 =
 
 
 R
 
 
 n
 
 
 
 ⊗
 
 
 R
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} ^{n}=\mathbb {R} ^{n}\otimes _{\mathbb {R} }\mathbb {C} }
 
 or instead as 
 
 
 
 
 
 C
 
 
 n
 
 
 =
 
 C
 
 
 ⊗
 
 
 R
 
 
 
 
 
 R
 
 
 n
 
 
 .
 
 
 {\displaystyle \mathbb {C} ^{n}=\mathbb {C} \otimes _{\mathbb {R} }\mathbb {R} ^{n}.}

If one orders the basis as 
 
 
 
 
 {
 
 
 e
 
 1
 
 
 ,
 i
 
 e
 
 1
 
 
 ,
 
 e
 
 2
 
 
 ,
 i
 
 e
 
 2
 
 
 ,
 …
 ,
 
 e
 
 n
 
 
 ,
 i
 
 e
 
 n
 
 
 
 }
 
 ,
 
 
 {\displaystyle \left\{e_{1},ie_{1},e_{2},ie_{2},\dots ,e_{n},ie_{n}\right\},}
 
 then the matrix for J takes the <a href="/facts/Block_diagonal/fPI8HX9f">block diagonal</a> form (subscripts added to indicate dimension):

J
          
            2
            n
          
        
        =
        
          
            [
            
              
                
                  0
                
                
                  −
                  1
                
              
              
                
                  1
                
                
                  0
                
              
              
                
                
                
                  0
                
                
                  −
                  1
                
              
              
                
                
                
                  1
                
                
                  0
                
              
              
                
                
                
                
                
                  ⋱
                
              
              
                
                
                
                
                
                
                  ⋱
                
              
              
                
                
                
                
                
                
                
                  0
                
                
                  −
                  1
                
              
              
                
                
                
                
                
                
                
                  1
                
                
                  0
                
              
            
            ]
          
        
        =
        
          
            [
            
              
                
                  
                    J
                    
                      2
                    
                  
                
              
              
                
                
                  
                    J
                    
                      2
                    
                  
                
              
              
                
                
                
                  ⋱
                
              
              
                
                
                
                
                  
                    J
                    
                      2
                    
                  
                
              
            
            ]
          
        
        .
      
    
    {\displaystyle J_{2n}={\begin{bmatrix}0&-1\\1&0\\&&0&-1\\&&1&0\\&&&&\ddots \\&&&&&\ddots \\&&&&&&0&-1\\&&&&&&1&0\end{bmatrix}}={\begin{bmatrix}J_{2}\\&J_{2}\\&&\ddots \\&&&J_{2}\end{bmatrix}}.}

This ordering has the advantage that it respects direct sums of complex vector spaces, meaning here that the basis for 
  
    
      
        
          
            C
          
          
            m
          
        
        ⊕
        
          
            C
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {C} ^{m}\oplus \mathbb {C} ^{n}}
  
 is the same as that for 
  
    
      
        
          
            C
          
          
            m
            +
            n
          
        
        .
      
    
    {\displaystyle \mathbb {C} ^{m+n}.}

On the other hand, if one orders the basis as 
 
 
 
 
 {
 
 
 e
 
 1
 
 
 ,
 
 e
 
 2
 
 
 ,
 …
 ,
 
 e
 
 n
 
 
 ,
 i
 
 e
 
 1
 
 
 ,
 i
 
 e
 
 2
 
 
 ,
 …
 ,
 i
 
 e
 
 n
 
 
 
 }
 
 
 
 {\displaystyle \left\{e_{1},e_{2},\dots ,e_{n},ie_{1},ie_{2},\dots ,ie_{n}\right\}}
 
, then the matrix for J is block-antidiagonal:

J
          
            2
            n
          
        
        =
        
          
            [
            
              
                
                  0
                
                
                  −
                  
                    I
                    
                      n
                    
                  
                
              
              
                
                  
                    I
                    
                      n
                    
                  
                
                
                  0
                
              
            
            ]
          
        
        .
      
    
    {\displaystyle J_{2n}={\begin{bmatrix}0&-I_{n}\\I_{n}&0\end{bmatrix}}.}

This ordering is more natural if one thinks of the complex space as a direct sum of real spaces, as discussed below.
The data of the real vector space and the J matrix is exactly the same as the data of the complex vector space, as the J matrix allows one to define complex multiplication. At the level of <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebras</a> and <a href="/facts/Lie_group/a9jCQyjF">Lie groups</a>, this corresponds to the inclusion of gl(n,C) in gl(2n,R) (Lie algebras – matrices, not necessarily invertible) and <a href="/facts/GL(n,C)/B54gNILS">GL(n,C)</a> in GL(2n,R):

gl(n,C) < gl(2n,R) and GL(n,C) < GL(2n,R).
The inclusion corresponds to forgetting the complex structure (and keeping only the real), while the subgroup GL(n,C) can be characterized (given in equations) as the matrices that commute with J:

G
          L
        
        (
        n
        ,
        
          C
        
        )
        =
        
          {
          
            A
            ∈
            
              G
              L
            
            (
            2
            n
            ,
            
              R
            
            )
            ∣
            A
            J
            =
            J
            A
          
          }
        
        .
      
    
    {\displaystyle \mathrm {GL} (n,\mathbb {C} )=\left\{A\in \mathrm {GL} (2n,\mathbb {R} )\mid AJ=JA\right\}.}

The corresponding statement about Lie algebras is that the subalgebra gl(n,C) of complex matrices are those whose <a href="/facts/Lie_bracket/n2gAUkNq">Lie bracket</a> with J vanishes, meaning 
 
 
 
 [
 J
 ,
 A
 ]
 =
 0
 ;
 
 
 {\displaystyle [J,A]=0;}
 
 in other words, as the kernel of the map of bracketing with J, 
 
 
 
 [
 J
 ,
 −
 ]
 .
 
 
 {\displaystyle [J,-].}

Note that the defining equations for these statements are the same, as 
 
 
 
 A
 J
 =
 J
 A
 
 
 {\displaystyle AJ=JA}
 
 is the same as 
 
 
 
 A
 J
 −
 J
 A
 =
 0
 ,
 
 
 {\displaystyle AJ-JA=0,}
 
 which is the same as 
 
 
 
 [
 A
 ,
 J
 ]
 =
 0
 ,
 
 
 {\displaystyle [A,J]=0,}
 
 though the meaning of the Lie bracket vanishing is less immediate geometrically than the meaning of commuting.

<h3>Direct sum</h3>
If V is any real vector space there is a canonical complex structure on the <a href="/facts/Direct_sum_of_vector_spaces/Y6PBJnjg">direct sum</a> V ⊕ V given by

J
        (
        v
        ,
        w
        )
        =
        (
        −
        w
        ,
        v
        )
        .
      
    
    {\displaystyle J(v,w)=(-w,v).}

The <a href="/facts/Block_matrix/fPI8HX9f">block matrix</a> form of J is

J
        =
        
          
            [
            
              
                
                  0
                
                
                  −
                  
                    I
                    
                      V
                    
                  
                
              
              
                
                  
                    I
                    
                      V
                    
                  
                
                
                  0
                
              
            
            ]
          
        
      
    
    {\displaystyle J={\begin{bmatrix}0&-I_{V}\\I_{V}&0\end{bmatrix}}}

where 
 
 
 
 
 I
 
 V
 
 
 
 
 {\displaystyle I_{V}}
 
 is the identity map on V. This corresponds to the complex structure on the tensor product 
 
 
 
 
 C
 
 
 ⊗
 
 
 R
 
 
 
 V
 .
 
 
 {\displaystyle \mathbb {C} \otimes _{\mathbb {R} }V.}

<h2 id="compatibility-with-other-structures">Compatibility with other structures</h2>
If B is a <a href="/facts/Bilinear_form/gyvzn9ym">bilinear form</a> on V then we say that J preserves B if

B
        (
        J
        u
        ,
        J
        v
        )
        =
        B
        (
        u
        ,
        v
        )
      
    
    {\displaystyle B(Ju,Jv)=B(u,v)}

for all u, v ∈ V. An equivalent characterization is that J is <a href="/facts/Skew-adjoint/wxjjnvCC">skew-adjoint</a> with respect to B:

B
        (
        J
        u
        ,
        v
        )
        =
        −
        B
        (
        u
        ,
        J
        v
        )
        .
      
    
    {\displaystyle B(Ju,v)=-B(u,Jv).}

If g is an <a href="/facts/Inner_product/HoyjElEy">inner product</a> on V then J preserves g if and only if J is an <a href="/facts/Orthogonal_transformation/s7LGBDFX">orthogonal transformation</a>. Likewise, J preserves a <a href="/facts/Nondegenerate/BqhqlXPH">nondegenerate</a>, <a href="/facts/Skew-symmetric_matrix/nw5CQeLN">skew-symmetric</a> form ω if and only if J is a <a href="/facts/Symplectic_transformation/kkWvzo5M">symplectic transformation</a> (that is, if 
 
 
 
 ω
 (
 J
 u
 ,
 J
 v
 )
 =
 ω
 (
 u
 ,
 v
 )
 
 
 {\textstyle \omega (Ju,Jv)=\omega (u,v)}
 
). For symplectic forms ω an interesting compatibility condition between J and ω is that

ω
        (
        u
        ,
        J
        u
        )
        >
        0
      
    
    {\displaystyle \omega (u,Ju)>0}

holds for all non-zero u in V. If this condition is satisfied, then we say that J tames ω (synonymously: that ω is tame with respect to J; that J is tame with respect to ω; or that the pair 
 
 
 
 (
 ω
 ,
 J
 )
 
 
 {\textstyle (\omega ,J)}
 
 is tame).
Given a symplectic form ω and a linear complex structure J on V, one may define an associated bilinear form gJ on V by

g
          
            J
          
        
        (
        u
        ,
        v
        )
        =
        ω
        (
        u
        ,
        J
        v
        )
        .
      
    
    {\displaystyle g_{J}(u,v)=\omega (u,Jv).}

Because a <a href="/facts/Symplectic_form/o95tWKBc">symplectic form</a> is nondegenerate, so is the associated bilinear form. The associated form is preserved by J if and only if the symplectic form is. Moreover, if the symplectic form is preserved by J, then the associated form is symmetric. If in addition ω is tamed by J, then the associated form is <a href="/facts/Definite_bilinear_form/utlBuVpN">positive definite</a>. Thus in this case V is an <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a> with respect to gJ.
If the symplectic form ω is preserved (but not necessarily tamed) by J, then gJ is the <a href="/facts/Complex_number/2w7mqI7e">real part</a> of the <a href="/facts/Hermitian_form/Kp6qUA8v">Hermitian form</a> (by convention antilinear in the first argument) 
 
 
 
 
 h
 
 J
 
 
 :
 
 V
 
 J
 
 
 ×
 
 V
 
 J
 
 
 →
 
 C
 
 
 
 {\textstyle h_{J}\colon V_{J}\times V_{J}\to \mathbb {C} }
 
 defined by

h
          
            J
          
        
        (
        u
        ,
        v
        )
        =
        
          g
          
            J
          
        
        (
        u
        ,
        v
        )
        +
        i
        
          g
          
            J
          
        
        (
        J
        u
        ,
        v
        )
        =
        ω
        (
        u
        ,
        J
        v
        )
        +
        i
        ω
        (
        u
        ,
        v
        )
        .
      
    
    {\displaystyle h_{J}(u,v)=g_{J}(u,v)+ig_{J}(Ju,v)=\omega (u,Jv)+i\omega (u,v).}

<h2 id="relation-to-complexifications">Relation to complexifications</h2>
Given any real vector space V we may define its <a href="/facts/Complexification/fDfFLccF">complexification</a> by <a href="/facts/Extension_of_scalars/RYGsTsYk">extension of scalars</a>:

V
          
            
              C
            
          
        
        =
        V
        
          ⊗
          
            
              R
            
          
        
        
          C
        
        .
      
    
    {\displaystyle 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 <a href="/facts/Complex_conjugation/7TEaoQDe">complex conjugation</a> defined by

v
              ⊗
              z
            
            ¯
          
        
        =
        v
        ⊗
        
          
            
              z
              ¯
            
          
        
      
    
    {\displaystyle {\overline {v\otimes z}}=v\otimes {\bar {z}}}

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

J
        (
        v
        ⊗
        z
        )
        =
        J
        (
        v
        )
        ⊗
        z
        .
      
    
    {\displaystyle J(v\otimes z)=J(v)\otimes z.}

Since C is <a href="/facts/Algebraically_closed/HuVQClok">algebraically closed</a>, J is guaranteed to have <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> which satisfy λ2 = −1, namely λ = ±i. Thus we may write

V
          
            
              C
            
          
        
        =
        
          V
          
            +
          
        
        ⊕
        
          V
          
            −
          
        
      
    
    {\displaystyle V^{\mathbb {C} }=V^{+}\oplus V^{-}}

where V+ and V− are the <a href="/facts/Eigenspace/8TjEoT8u">eigenspaces</a> of +i and −i, respectively. Complex conjugation interchanges V+ and V−. The projection maps onto the V± eigenspaces are given by

P
            
          
          
            ±
          
        
        =
        
          
            1
            2
          
        
        (
        1
        ∓
        i
        J
        )
        .
      
    
    {\displaystyle {\mathcal {P}}^{\pm }={1 \over 2}(1\mp iJ).}

So that

V
          
            ±
          
        
        =
        {
        v
        ⊗
        1
        ∓
        J
        v
        ⊗
        i
        :
        v
        ∈
        V
        }
        .
      
    
    {\displaystyle V^{\pm }=\{v\otimes 1\mp Jv\otimes 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 <a href="/facts/Complex_conjugate_vector_space/trWp8y4A">complex conjugate</a> 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
          
            
              C
            
          
        
        ≅
        W
        ⊕
        
          
            W
            ¯
          
        
        .
      
    
    {\displaystyle W^{\mathbb {C} }\cong W\oplus {\overline {W}}.}

<h2 id="extension-to-related-vector-spaces">Extension to related vector spaces</h2>
Let V be a real vector space with a complex structure J. The <a href="/facts/Dual_space/4Uw4Knz1">dual space</a> V* has a natural complex structure J* given by the dual (or <a href="/facts/Transpose/8wmsagGS">transpose</a>) of J. The complexification of the dual space (V*)C therefore has a natural decomposition

(
        
          V
          
            ∗
          
        
        
          )
          
            
              C
            
          
        
        =
        (
        
          V
          
            ∗
          
        
        
          )
          
            +
          
        
        ⊕
        (
        
          V
          
            ∗
          
        
        
          )
          
            −
          
        
      
    
    {\displaystyle (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) <a href="/facts/Tensor_algebra/H7CKKryf">tensor</a>, <a href="/facts/Symmetric_algebra/A8FthY0N">symmetric</a>, and <a href="/facts/Exterior_algebra/rdN4TvpR">exterior algebras</a> 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:

Λ
          
            r
          
        
        U
        =
        
          ⨁
          
            p
            +
            q
            =
            r
          
        
        (
        
          Λ
          
            p
          
        
        S
        )
        ⊗
        (
        
          Λ
          
            q
          
        
        T
        )
        .
      
    
    {\displaystyle \Lambda ^{r}U=\bigoplus _{p+q=r}(\Lambda ^{p}S)\otimes (\Lambda ^{q}T).}

A complex structure J on V therefore induces a decomposition

Λ
          
            r
          
        
        
        
          V
          
            
              C
            
          
        
        =
        
          ⨁
          
            p
            +
            q
            =
            r
          
        
        
          Λ
          
            p
            ,
            q
          
        
        
        
          V
          
            J
          
        
      
    
    {\displaystyle \Lambda ^{r}\,V^{\mathbb {C} }=\bigoplus _{p+q=r}\Lambda ^{p,q}\,V_{J}}

where

Λ
          
            p
            ,
            q
          
        
        
        
          V
          
            J
          
        
        
        
          
            
              
                =
              
              
                
                  d
                  e
                  f
                
              
            
          
        
        
        (
        
          Λ
          
            p
          
        
        
        
          V
          
            +
          
        
        )
        ⊗
        (
        
          Λ
          
            q
          
        
        
        
          V
          
            −
          
        
        )
        .
      
    
    {\displaystyle \Lambda ^{p,q}\,V_{J}\;{\stackrel {\mathrm {def} }{=}}\,(\Lambda ^{p}\,V^{+})\otimes (\Lambda ^{q}\,V^{-}).}

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

dim
          
            
              C
            
          
        
        ⁡
        
          Λ
          
            r
          
        
        
        
          V
          
            
              C
            
          
        
        =
        
          
            
              (
            
            
              
                2
                n
              
              r
            
            
              )
            
          
        
        
        
          dim
          
            
              C
            
          
        
        ⁡
        
          Λ
          
            p
            ,
            q
          
        
        
        
          V
          
            J
          
        
        =
        
          
            
              (
            
            
              n
              p
            
            
              )
            
          
        
        
          
            
              (
            
            
              n
              q
            
            
              )
            
          
        
        .
      
    
    {\displaystyle \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 <a href="/facts/Vandermonde%27s_identity/pvm6ubrH">Vandermonde's identity</a>.
The space of (p,q)-forms Λp,q VJ* is the space of (complex) <a href="/facts/Multilinear_form/Rk420AcK">multilinear forms</a> 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 <a href="/facts/Multilinear_map/YosSfBKE">multilinear maps</a> from VJ to C which are complex linear in p terms and <a href="/facts/Conjugate-linear/GTXzWMKc">conjugate-linear</a> in q terms.
See <a href="/facts/Complex_differential_form/307vM8pd">complex differential form</a> and <a href="/facts/Almost_complex_manifold/wsXTsuC6">almost complex manifold</a> for applications of these ideas.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Almost_complex_manifold/wsXTsuC6">Almost complex manifold</a></li>
<li><a href="/facts/Complex_manifold/DlvPnpzm">Complex manifold</a></li>
<li><a href="/facts/Complex_differential_form/307vM8pd">Complex differential form</a></li>
<li><a href="/facts/Complex_conjugate_vector_space/trWp8y4A">Complex conjugate vector space</a></li>
<li><a href="/facts/Hermitian_structure/69fYok0x">Hermitian structure</a></li>
<li><a href="/facts/Real_structure/vahYuPa6">Real structure</a></li></ul>

<ul><li>Kobayashi S. and Nomizu K., <a href="/facts/Foundations_of_Differential_Geometry/lgGBjhhd">Foundations of Differential Geometry</a>, John Wiley & Sons, 1969. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-470-49648-7. (complex structures are discussed in Volume II, Chapter IX, section 1).</li>
<li>Budinich, P. and Trautman, A. The Spinorial Chessboard, Springer-Verlag, 1988. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-19078-3. (complex structures are discussed in section 3.1).</li>
<li>Goldberg S.I., Curvature and Homology, Dover Publications, 1982. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-64314-X. (complex structures and almost complex manifolds are discussed in section 5.2).</li></ul>

Linear complex structure open-in-new

Linear complex structure