What is the Difference Between a Lie Subalgebra and a Subspace?

[KarateMan]

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.

 

[mrandersdk]

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]

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]

an algebra is nececearily not a space

This is terribly, terribly wrong.

 

[mrandersdk]

so sorry always do this, i read it as lie group, why do i always do this. Sorry again.

Neglect my comment.

 

[KarateMan]

Thanks everyone. took me a while but I think I swallowed it!