What Does Closed Mean in the Context of Lie Subgroups?

[malawi_glenn]

Hi, I was reading Cartan's Theorem:

A Group H is a Lie Subgroup to Lie Group G if H is a closed subgroup to G.

Now first of all, is this a definition of Lie Subgroup?

Second, what does it mean that the subgroup is "closed"? I thought all groups where closed under group multiplication.. :/ Help?

 

[bartadam]

A group is a set G with elements (possible infinite and/or uncountable) such as

$$G= \left\{g_1,g_2,\cdots g_n\right\}$$

with the following

1) a multiplication rule $$g_i*g_j=g_k$$ for some i,j and k. (closure)
2) an identity element I, st $$I*g_i=g_i*I=g_i$$
3) every element has an inverse, i.e $$g_i*g_i^{-1}=I$$

A SUBSET H is a subgroup if $$h_i*h_j=h_k$$ for every element h in the subset H

 

[malawi_glenn]

yes i know, but what is meant by a CLOSED subgroup?!

you forgot associativity by the way...


and that the inverse to h is in H ...

 

[bartadam]

I don't understand what the problem is.

All subgroups are closed otherwise it's not a subgroup it's just merely some subset of G

 

[malawi_glenn]

closed with respect to what? group multiplication?

Why is the theorem stressing CLOSED subgroup, the word has no power if there are no subgroups which are NOT CLOSED (open)..

 

[bartadam]

yes group multiplication.

I don't know why it's specifying closed subgroup to be honest. The word is redundent.

Yes I didn't bother with associativity but at this time in the evening I couldn't be arsed lol

 

[d_leet]

I'm fairly certain it means closed in the topological sense as a subset of the Lie group G.

 

[cristo]

I don't think it means closed as in closed under group multiplication since (as both of you say) that is always true in the definition of a group. What the statement probably means is that H is a Lie subgroup of G if H is a (topologically) closed subgroup of G.

 

[malawi_glenn]

Ok, so a subgroup of G which is also a closed subset of G?

 

[malawi_glenn]

I don't think it means closed as in closed under group multiplication since (as both of you say) that is always true in the definition of a group. What the statement probably means is that H is a Lie subgroup of G if H is a (topologically) closed subgroup of G.

So how for instance do one determine if a group is topologically closed? For example SO(2), is that a closed group?

 

[samalkhaiat]

A Group H is a Lie Subgroup to Lie Group G if H is a closed subgroup to G.

The word Lie need not be there.

Now first of all, is this a definition of Lie Subgroup?

Yes, but the wording is bad. This is why you are asking about "closed".

Second, what does it mean that the subgroup is "closed"? I thought all groups where closed under group multiplication.. :/ Help?

A subgroup H of a group G is a subset {h} of elements of G which closes with respect to the multiplication already defined by G ( h.k in H for all h,k in H) and which contains the inverse of each of its elements h, and the identity e. If the subset {h} does not close, i.e., if

$$h.k \in G \ \mbox{not in H}, \ h,k \in H$$

then the set {h} does form a group.

Subgroups arise by imposing a restriction on the original group, e.g. restricting the 3-dimensional group of rotations to rotations about one axis, or to discrete rotations.
The most interesting subgroups are the so-called invariant subgroups; an invariant subgroup H of G is one which is invariant with respect to conjugation with G, i.e.,

$$ghg^{-1} \in H \ \ \mbox{for all} \ \ h \in H, \ g \in G$$

The translation subgroups of the spacetime groups are invariant subgroups because they are transformed into themselves by rotations.
Also, when H is an invariant subgroup, the coset manifold G/H forms a group called the quotient group.

regards

sam

 

[malawi_glenn]

The word Lie need not be there.



Yes, but the wording is bad. This is why you are asking about "closed".

But I am asking about What a Lie Subgroup is. What is it?

 

[samalkhaiat]

But I am asking about What a Lie Subgroup is. What is it?

Add the word Lie before the words group and subgroup in the my post!

 

[malawi_glenn]

so,

a subgroup to a Lie group is a "Lie subgroup" ?

Now SO(2) is a liegroup, is U(1) a Lie Sub group to SO(2)?

 

[samalkhaiat]

a subgroup to a Lie group is a "Lie subgroup" ?

Yes.

Now SO(2) is a liegroup, is U(1) a Lie Sub group to SO(2)?

No. they are isomorphic.

U(1) is a subgroup of SU(2) and SO(2) is a subgroup of SO(3).

 

[malawi_glenn]

ok, and since SO(3) is a Lie Group, SO(2) is a Lie Subgroup to SO(3)?

 

[samalkhaiat]

ok, and since SO(3) is a Lie Group, SO(2) is a Lie Subgroup to SO(3)?

the group G = {e} of the identity, is a subgroup of any group A. If A is a Lie group, would we call {e} a Lie subgroup? Yes, it is a trivial Lie group.

 

[malawi_glenn]

Ok I think I got it!

Really hard for a physicsists to get involved into abstract algebra, but I try hard to understand.

Thanx for all replies!

 

[morphism]

so,

a subgroup to a Lie group is a "Lie subgroup" ?

No. Presumably we want a "Lie subgroup" to be a Lie group itself. Now, any subgroup H of a Lie group G can inherit the topological structure of G (as a subspace of G) - so at least there is a topology on H that we can talk about - but H may not inherit the manifold structure associated to G. In other words, H may not be a submanifold of G. Example: the real numbers R form a Lie group under addition, the usual topology, and the usual smooth structure. However, the subgroup Q of rational numbers is not a manifold, so it's not a Lie group. (Note that Q is not topologically closed in R.) The actual definition of a Lie subgroup of a Lie group varies a bit, but a common one is the following: a subgroup H of a Lie group G is a Lie subgroup of G if H is an embedded submanifold of G. Check your text to see what definition the author is using.

Now, it's a standard result that a Lie subgroup of a Lie group G is topologically closed in G. Cartan's theorem states that the converse of this result is true (at least for real Lie groups!).

 

[malawi_glenn]

Ok I have no book but is using different internet resources.

So let me get this straight, A Lie Subgroup is a subgroup to a Lie Group which is a Lie Group itself?

 

[morphism]

Using the internet as your primary source for learning about Lie groups is a bad idea. Just get a textbook. Plenty of good ones out there.

And no, a Lie subgroup H of a Lie group G is not 'just' a subgroup which is a Lie group itself. The Lie group structure of H should be related to that of G.

 

[malawi_glenn]

I meant I have textbooks from internet, such as lecture notes etc.

So what is Lie Group structure referring to in this case?

Is the Special Linear Group a Lie subgroup to the General Linear group, for instance.?

SL is both a subgroup of GL, and SL is also a Lie group. But what is meant by "structure should be related to that of GL" ?...

 

[morphism]

I was specific in my first post: H has to be an embedded submanifold of G. This ensures that H's topological and manifold structure is related to that of G.

SL(n) is closed in GL(n), so yes, it's a Lie subgroup of GL(n). SL(n) inherits the subspace topology from GL(n), and it's a smooth submanifold of GL(n) in a very natural way (thanks to the smoothness of the determinant).

 

[malawi_glenn]

Thank you, you have clearly explained this well to me!

Now, the problem with books is that I found so many which seems to be good, but also expensive. Thats why I am using online material och tutorials.

If you would recommend me, as a theoretical physicsits, one book - which?

Cheers

 

[morphism]

To be honest, I don't know what I would recommend to a physicist. Maybe you could try asking in one of the physics forums (or the book forum).

 

[malawi_glenn]

To be honest, I don't know what I would recommend to a physicist. Maybe you could try asking in one of the physics forums (or the book forum).

I appreciate your honesty and your guidance so far! :-)

Will ask there, and also try to ask some teacher at my university, they should know what fits me.

 

[morphism]

No problem!