Lie subalgebra and subspace


by KarateMan
Tags: subalgebra, subspace
KarateMan
KarateMan is offline
#1
Mar12-08, 01:39 AM
P: 13
I have a question about Lie subalgebra.

They say "a Lie subalgebra is a much more CONSTRAINED structure than a subspace".
Well, it seems subtle, and I find this very tricky to follow.

Can anyone explain this with concrete examples?

If my question is not clear, please tell me so, I will try to rephrase it.
Thanks.
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mrandersdk
mrandersdk is offline
#2
Mar12-08, 07:44 AM
P: 230
an algebra is nececearily not a space (understood vectorspace), so there is a big different. If you are talking about an subalgebra and a lie subalgebra. I guess you know the usual deffinition of a subalgebra, a lie subalgebra is a much more strict because a lie subalgebra needs to be a algebra + a submanifold, which is very strict.
ichbinfrodo
ichbinfrodo is offline
#3
Mar12-08, 12:51 PM
P: 5
KarateMan: A Lie subalgebra is a linear subspace which is a Lie algebra.
Hence, besides being a subspace, it has to satisfy the Lie algebra axioms (e.g. it has to be closed under the Lie bracket!).

mrandersdk: There are no topological requirements for Lie (sub)algebras.

shoehorn
shoehorn is offline
#4
Mar12-08, 01:34 PM
P: 448

Lie subalgebra and subspace


Quote Quote by mrandersdk View Post
an algebra is nececearily not a space
This is terribly, terribly wrong.
mrandersdk
mrandersdk is offline
#5
Mar13-08, 02:02 AM
P: 230
so sorry always do this, i read it as lie group, why do i always do this. Sorry again.

Neglect my comment.
KarateMan
KarateMan is offline
#6
Mar16-08, 09:53 PM
P: 13
Thanks everyone. took me a while but I think I swallowed it!


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