Editing Linear complex structure (section)

=== Complex ''n''-dimensional space '''C'''<sup>''n''</sup> ===
The fundamental example of a linear complex structure is the structure on '''R'''<sup>2''n''</sup> coming from the complex structure on '''C'''<sup>''n''</sup>. That is, the complex ''n''-dimensional space '''C'''<sup>''n''</sup> is also a real 2''n''-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  <math> i (\lambda v) = (i \lambda) v = (\lambda i) v = \lambda (i v) </math> – and distributes across vector addition. As a complex ''n''×''n'' matrix, this is simply the [[scalar matrix]] with ''i'' on the diagonal. The corresponding real 2''n''×2''n'' matrix is denoted ''J''.

Given a basis <math>\left\{e_1, e_2, \dots, e_n \right\}</math> for the complex space, this set, together with these vectors multiplied by ''i,'' namely <math>\left\{ie_1, ie_2, \dots, ie_n\right\},</math> 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 <math>\Complex^n = \R^n \otimes_{\R} \Complex</math> or instead as <math>\Complex^n = \Complex \otimes_{\R} \R^n.</math>

If one orders the basis as <math>\left\{e_1, ie_1, e_2, ie_2, \dots, e_n, ie_n\right\},</math> then the matrix for ''J'' takes the [[block diagonal]] form (subscripts added to indicate dimension):
<math display="block">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}.</math>
This ordering has the advantage that it respects direct sums of complex vector spaces, meaning here that the basis for <math>\Complex^m \oplus \Complex^n</math> is the same as that for <math>\Complex^{m+n}.</math>

On the other hand, if one orders the basis as <math>\left\{e_1,e_2,\dots,e_n, ie_1, ie_2, \dots, ie_n\right\}</math>, then the matrix for ''J'' is block-antidiagonal:
<math display="block">J_{2n} = \begin{bmatrix}0 & -I_n \\ I_n & 0\end{bmatrix}.</math>
This ordering is more natural if one thinks of the complex space as a [[#Direct sum|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 [[Lie algebra]]s and [[Lie group]]s, this corresponds to the inclusion of gl(''n'','''C''') in gl(2''n'','''R''') (Lie algebras – matrices, not necessarily invertible) and [[GL(n,C)|GL(''n'','''C''')]] in GL(2''n'','''R'''):
{{block indent | em = 1.5 | text = 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:''
<math display="block">\mathrm{GL}(n, \Complex) = \left\{ A \in \mathrm{GL}(2n,\R) \mid AJ = JA \right\}.</math>
The corresponding statement about Lie algebras is that the subalgebra gl(''n'','''C''') of complex matrices are those whose [[Lie bracket]] with ''J'' vanishes, meaning <math>[J,A] = 0;</math> in other words, as the kernel of the map of bracketing with ''J,'' <math>[J,-].</math>

Note that the defining equations for these statements are the same, as <math>AJ = JA</math> is the same as <math>AJ - JA = 0,</math> which is the same as <math>[A,J] = 0,</math> though the meaning of the Lie bracket vanishing is less immediate geometrically than the meaning of commuting.