f******h
发帖数: 104 | 1 Prove that PGL(n) is isomorphic to a closed subgroup of GL(n^2),
where PGL(n) is the projective linear group.
Thanks. |
l******e
发帖数: 4 | 2 Consider the conjugation action of GL(n) on the space M(n) of n by n matrices. This defines a representation of GL(n) in M(n) whose kernel is just the scalars in GL(n).
【在 f******h 的大作中提到】
: Prove that PGL(n) is isomorphic to a closed subgroup of GL(n^2),
: where PGL(n) is the projective linear group.
: Thanks.
|
f******h
发帖数: 104 | 3 Thank you. I have one more question:
how can you prove that the induced map
on regular functions
k[GL(n^2)] --> k[GL(n)] is surjective?
I feel that otherwise the adjoint
representation may be not a closed
immersion.(although the image is closed)
Sorry for my stupidity:)
matrices. This defines a representation of GL(n) in M(n) whose kernel is
just the scalars in GL(n).
【在 l******e 的大作中提到】
: Consider the conjugation action of GL(n) on the space M(n) of n by n matrices. This defines a representation of GL(n) in M(n) whose kernel is just the scalars in GL(n).
|
l******e
发帖数: 4 | 4 We don't need to show that k[GL(n^2)] --> k[PGL(n)] is surjective. A well-
known theorem asserts that if f:G->H is a surjective homomorphism of
algebraic groups, then it is a quotient homomorphism iff df is surjective.
In our case, the differential of GL(n)->GL(n^2) is just the adjoint
representation of gl(n), which has a 1-dimensional kernel.
【在 f******h 的大作中提到】
: Thank you. I have one more question:
: how can you prove that the induced map
: on regular functions
: k[GL(n^2)] --> k[GL(n)] is surjective?
: I feel that otherwise the adjoint
: representation may be not a closed
: immersion.(although the image is closed)
: Sorry for my stupidity:)
:
: matrices. This defines a representation of GL(n) in M(n) whose kernel is
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