# How to construct an isomorphism between the Complexified Special Linear Lie Group and the Special Unitary Group?

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This may be an unenlightening question, but I'm just not sure about the result and hoping someone can help me varify it.

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This question is related to these three questions.

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I want to construct the isomorphism relationship between the Lie Groups $SL(2,\mathbb{C})$ and $SU(2)$. I have the feeling that there should be some such isomorphism of groups.

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To begin, we know that as Lie Algebras

$$\mathfrak{sl}(2,\mathbb{C}) \simeq \mathfrak{so}(1,3)$$

and

$$\mathfrak{su}(2) \oplus \mathfrak{su}(2) \simeq \mathfrak{o}(4)$$

But we also know that

$$\mathfrak{so}(n) \simeq \mathfrak{o}(n)$$

so I believe that this allows us to write

$$\mathfrak{su}(2) \oplus \mathfrak{su}(2) \simeq \mathfrak{sl}(2,\mathbb{C})$$

This makes sense anyway, since we know that the real algebra of the complexification of $\mathfrak{su}(2)$ is $\mathfrak{sl}(2,\mathbb{C})$, 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 $G_1$, $G_2$ be Lie groups. Then $G_1$ and $G_2$ 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, $SL(2,\mathbb{C})$ 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 $SO(1,3)\uparrow$ and $SO(3)$ respectively, and so by the definition they must be simply connected.

This would solve our problem, and we could write down

$$SU(2) \times SU(2) \simeq SL(2,\mathbb{C})$$

and be done.

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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).

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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.

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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).

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This post imported from StackExchange Physics at 2014-07-06 07:53 (UCT), posted by SE-user Flint72
Comment to the question (v1): The two Lie groups $G_1=SL(2,\mathbb{C})$ and $G_2=SU(2)\times SU(2)$ are not isomorphic. For starters, $G_1$ is non-compact and $G_2$ is compact.

This post imported from StackExchange Physics at 2014-07-06 07:53 (UCT), posted by SE-user Qmechanic
Ah, ok, so then we certainly won't be able to find an isomorphism of groups between them! Is what I have written correct, if pointless, all the same? Namely, am I indeed correct in thinking that as Lie algebras $$\mathfrak{su}(2) \oplus \mathfrak{su}(2) \simeq \mathfrak{sl}(2,\mathbb{C})$$ ? Finally, does this mean that the best we can do is have a local isomorphism, as per that theorem of Lie? Thank you!

This post imported from StackExchange Physics at 2014-07-06 07:53 (UCT), posted by SE-user Flint72
The two real Lie algebras ${\rm Lie}(G_1)=sl(2,\mathbb{C})\cong so(1,3;\mathbb{R})$ and ${\rm Lie}(G_2)=su(2) \oplus su(2)\cong so(4;\mathbb{R})$ are not isomorphic.

This post imported from StackExchange Physics at 2014-07-06 07:53 (UCT), posted by SE-user Qmechanic
@Qmechanic : Ah, I see. So this is obviously where I have gone. I notice that you have said that the real Lie albegras $G_1$ and $G_2$ are not isomorphic. Is this still the case for their complixifications? I will edit the question to take account of these points in due corse. Though I think now there is perhaps no saving this question? Thanks again.

This post imported from StackExchange Physics at 2014-07-06 07:53 (UCT), posted by SE-user Flint72
Yes, the complexification is $sl(2,\mathbb{C})\oplus sl(2,\mathbb{C})$ for both Lie algebras.

This post imported from StackExchange Physics at 2014-07-06 07:53 (UCT), posted by SE-user Qmechanic
SL(2, C) and SU(2) are not isomorphic, but there can be some homomorphism between the 2

This post imported from StackExchange Physics at 2014-07-06 07:53 (UCT), posted by SE-user Nikos M.

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