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Concept
Linear complex structure
Résumé
In mathematics, a complex structure on a real vector space V is an automorphism of V that 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.
Every complex vector space can be equipped with a compatible complex structure, however, there is in general no canonical such structure. 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, by contrast to complex manifolds. The term "complex structure" often refers to this structure on manifolds; when it refers instead to a structure on vector spaces, it may be called a linear complex structure.
A complex structure on a real vector space V is a real linear transformation
such that
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
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 VJ.
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 = iw for all w ∈ 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 i2 = −1. Then a representation of C is a real vector space V, together with an action of C on V (a map C → End(V)). 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.
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