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 4 replaced http://physics.stackexchange.com/ with https://physics.stackexchange.com/ edited Apr 13 '17 at 12:39 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 thesethese threethree questionsquestions. $$\\$$ 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. $$\\$$ 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. $$\\$$ 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 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 $$SL(2,\mathbb{C})$$ and $$SU(2)$$. I have the feeling that there should be some such isomorphism of groups. $$\\$$ To begin, we know that as Lie Algebras $$\mathfrak{sl}(2,\mathbb{C}) \simeq \mathfrak{so}(1,3)$$ 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. $$\\$$ 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 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 $$SL(2,\mathbb{C})$$ and $$SU(2)$$. I have the feeling that there should be some such isomorphism of groups. $$\\$$ To begin, we know that as Lie Algebras $$\mathfrak{sl}(2,\mathbb{C}) \simeq \mathfrak{so}(1,3)$$ 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. $$\\$$ 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). $$\\$$ Post Closed as "duplicate" by Qmechanic♦ of occurred Aug 20 '15 at 9:47 Tweeted twitter.com/#!/StackPhysics/status/492605856044048384 occurred Jul 25 '14 at 9:42 3 added the maths-physics tag | link edited May 21 '14 at 14:35 Flint72 1,27877 silver badges2626 bronze badges 2 removed thanks edited May 21 '14 at 14:12 Qmechanic♦ 115k1414 gold badges229229 silver badges13711371 bronze badges 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 thesethese threethree questionsquestions. $$\\$$ 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. $$\\$$ 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 replationshiprelationship 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. $$\\$$ However I want to try to varifyverify 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 varifyverify 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). $$\\$$ Thanks! 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 $$SL(2,\mathbb{C})$$ and $$SU(2)$$. I have the feeling that there should be some such isomorphism of groups. $$\\$$ To begin, we know that as Lie Algebras $$\mathfrak{sl}(2,\mathbb{C}) \simeq \mathfrak{so}(1,3)$$ 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 replationship 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. $$\\$$ However I want to try to varify 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 varify 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). $$\\$$ Thanks! 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 $$SL(2,\mathbb{C})$$ and $$SU(2)$$. I have the feeling that there should be some such isomorphism of groups. $$\\$$ To begin, we know that as Lie Algebras $$\mathfrak{sl}(2,\mathbb{C}) \simeq \mathfrak{so}(1,3)$$ 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. $$\\$$ 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). $$\\$$ 1 asked May 21 '14 at 14:06 Flint72 1,27877 silver badges2626 bronze badges