I am confused about the following paragraph in Fulton W., Harris J. "Representation Theory: A First Course":
Of course, the group $GL_n\mathbb{C}$ of complex linear automorphisms
of a complex vector space $V=\mathbb{C}^n$ can be viewed as subgroup
of the general linear group $GL_{2n}\mathbb{R};$ it is, thus, a real
Lie group as well, as is the subgroup $SL_n\mathbb{C}$. Similarly, the
subgroups $SO_n\mathbb{C} \subset SL_n\mathbb{C}$ and
$Sp_{2n}\mathbb{C} \subset SL_{2n}\mathbb{C}$ are real as well as
complex Lie subgroups. Note that since all nondegenerate bilinear
symmetric forms on a complex vector space are isomorphic (in
particular, there is no such thing as a signature), there is one
complex orthogonal subgroup $SO_n\mathbb{C} \subset SL_n\mathbb{C}$ up
to conjugation; there are no analogs of the groups
$SO_{k,l}\mathbb{R}.$
The things I don't understand:
- How do we get the subgroup structure on $GL_n\mathbb{C}$? It seems, that according to the definition of a subgroup (it is also a closed submanifold), we should construct a map $f: GL_n\mathbb{C} \rightarrow GL_{2n}\mathbb{R}$. Should $f$ also be a homomorphism of groups? If it is the case, then I can hardly imagine its coordinate representation.
- How do we prove that all nondegenerate bilinear symmetric forms are isomorphic?
- I have never encountered such a designation $SO_n\mathbb{C}$, it was always $SO_n$. And now I don't understand why we ignore the field.
Also, I want to emphasize that before this paragraph, the structure of a complex manifold was not defined.
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