Linear complex structure
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
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