This may be an unenlightening question, but I'm just not sure about the result and hoping someone can help me varify it.
This question is related to these three questions.
I want to construct the isomorphism relationship between the Lie Groups SL(2,C) and SU(2). I have the feeling that there should be some such isomorphism of groups.
To begin, we know that as Lie Algebras
sl(2,C)≃so(1,3)
and
su(2)⊕su(2)≃o(4)
But we also know that
so(n)≃o(n)
so I believe that this allows us to write
su(2)⊕su(2)≃sl(2,C)
This makes sense anyway, since we know that the real algebra of the complexification of su(2) is sl(2,C), and in taking the real algebra of the complexified Lie algebra we get two commuting copies.
So, the part that I am not yet convinced about is how to get from this relationship between algebras to a relationship between groups.
I was told by someone in the department that
Theorem The Fundamental Theorem of lie Groups: Let G1, G2 be Lie groups. Then G1 and G2 have isomorphic Lie algebras if and only if they are locally isomorphic.
So this is a local statement only.
Moreover, he said that there is an extension of this theorem to a global statement which says that the Lie groups are globally isomorphic if they are simply connected.
Now, for our two groups, SL(2,C) and SU(2), we know that they are indeed simply connected. We could prove this, or instead, recall that they are the Universal Covering Groups of SO(1,3)↑ and SO(3) respectively, and so by the definition they must be simply connected.
This would solve our problem, and we could write down
SU(2)×SU(2)≃SL(2,C)
and be done.
However I want to try to verify that statement, as opposed to taking it in blind faith (not that I have any reason to doubt it, but rather that I'd like to 'learn it' as opposed 'to be aware of it', if that makes sense).
I tried looking it up, and the obvious source didn't have anything on a Fundamental Theorem of Lie Groups, only a short bit on The Third Theorem of Lie.
Some searching brought up these lecture notes (in .pdf format) from UCLA. It appears to be getting at what I want, but unfortunately is written in category theoretic language, which I know nothing about.
Could anyone verify for me if this is correct, and perhaps point me to a book/ website/ lecture notes etc. where I could reference. (Our library is huge, so a book being online need not be a constraint).
This post imported from StackExchange Physics at 2014-07-06 07:53 (UCT), posted by SE-user Flint72