Linear Complex Structure - Definition and Properties

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.

More formally, a linear complex structure on a real vector space is an algebra representation of the complex numbers C, thought of as an associative algebra over the real numbers. This algebra is realized concretely as which corresponds to Then a representation of C is a real vector space V, together with an action of C on V (a map ). Concretely, this is just an action of i, as this generates the algebra, and the operator representing i (the image of i in End(V)) is exactly J.

If V _J has complex dimension n then V must have real dimension 2n. That is, a finite-dimensional space 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 is a basis for the complex vector space V _J then 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