Editing Linear complex structure (section)

==Definition and properties==

A '''complex structure''' on a [[real vector space]] <math>V</math> is a real [[linear transformation]]
<math display=block>J :V \to V</math>
such that
<math display=block>J^2 = -\text{id}_V.</math>
Here <math>J^2</math> means <math>J</math> [[function composition|composed]] with itself and <math>\text{id}_V</math> is the [[identity function|identity map]] on <math>V</math>. That is, the effect of applying <math>J</math> twice is the same as multiplication by <math>-1</math>. This is reminiscent of multiplication by the [[imaginary unit]], <math>i</math>. A complex structure allows one to endow <math>V</math> with the structure of a [[complex vector space]]. Complex scalar multiplication can be defined by
<math display=block>(x + iy)\vec{v} = x\vec{v} + yJ(\vec{v})</math>
for all real numbers <math>x,y</math> and all vectors <math>\vec{v}</math> in {{math|''V''}}. One can check that this does, in fact, give <math>V</math> the structure of a complex vector space which we denote <math>V_J</math>.

Going in the other direction, if one starts with a complex vector space <math>W</math> then one can define a complex structure on the underlying real space by defining <math>Jw = iw~~\forall w\in W</math>.

More formally, a linear complex structure on a real vector space is an [[algebra representation]] of the [[complex number]]s <math>\mathbb{C}</math>, thought of as an [[associative algebra]] over the [[real number]]s. This algebra is realized concretely as 
<math display=block>\Complex = \Reals[x]/(x^2+1),</math>
which corresponds to <math>i^2=-1</math>. Then a representation of <math>\mathbb{C}</math> is a real vector space <math>V</math>, together with an action of <math>\mathbb{C}</math> on <math>V</math> (a map <math>\mathbb{C}\rightarrow \text{End}(V)</math>). Concretely, this is just an action of <math>i</math>, as this generates the algebra, and the operator representing <math>i</math> (the image of <math>i</math> in <math>\text{End}(V)</math>) is exactly <math>J</math>.

If <math>V_J</math> has complex [[dimension (linear algebra)|dimension]] <math>n</math>, then <math>V</math> must have real dimension <math>2n</math>. That is, a finite-dimensional space <math>V</math> 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 <math>J</math> on pairs <math>e,f</math> of [[basis (linear algebra)|basis]] vectors by <math>Je=f</math> and <math>Jf=-e</math> and then [[extend by linearity]] to all of <math>V</math>. If <math>(v_1, \dots,v_n)</math> is a basis for the complex vector space <math>V_J</math> then <math>(v_1,Jv_1,\dots ,v_n ,Jv_n)</math> is a basis for the underlying real space <math>V</math>.

A real linear transformation <math>A:V \rightarrow V</math> is a '''''complex''''' linear transformation of the corresponding complex space <math>V_J</math> [[if and only if]] <math>A</math> commutes with <math>J</math>, i.e. if and only if
<math display=block>AJ = JA.</math>
Likewise, a real [[Linear subspace|subspace]] <math>U</math> of <math>V</math> is a complex subspace of <math>V_J</math> if and only if <math>J</math> preserves <math>U</math>, i.e. if and only if
<math display=block>JU = U.</math>