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

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In summary, a Lie subalgebra is a linear subspace that also satisfies the Lie algebra axioms, making it more constrained than a regular subspace. Unlike a subspace, it must also be a submanifold and be closed under the Lie bracket. There are no topological requirements for Lie subalgebras.
  • #1
KarateMan
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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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  • #2
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.
 
  • #3
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.
 
  • #4
mrandersdk said:
an algebra is nececearily not a space

This is terribly, terribly wrong.
 
  • #5
so sorry always do this, i read it as lie group, why do i always do this. Sorry again.

Neglect my comment.
 
  • #6
Thanks everyone. took me a while but I think I swallowed it!
 

1. What is a Lie subalgebra?

A Lie subalgebra is a subset of a Lie algebra that is closed under the Lie bracket operation. This means that the Lie bracket of any two elements in the subalgebra is also an element of the subalgebra.

2. What is the significance of Lie subalgebras?

Lie subalgebras play a crucial role in the study of Lie groups and their representations. They provide a way to break down a complex Lie algebra into smaller, more manageable subalgebras.

3. How is a Lie subalgebra related to a subspace?

A Lie subalgebra is a type of subspace, as it is a subset of a larger vector space that is closed under a specific operation (the Lie bracket). However, not all subspaces are Lie subalgebras, as they may not satisfy the closure property.

4. Can a Lie subalgebra be a subalgebra of a different Lie algebra?

Yes, a Lie subalgebra can be a subset of a different Lie algebra. In fact, this is often the case when studying the relationships between different Lie algebras and their corresponding Lie groups.

5. How are Lie subalgebras used in physics?

Lie subalgebras are used in physics to describe the symmetries and transformations of physical systems. They are particularly important in the study of gauge theories, which are used to describe fundamental interactions in particle physics.

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