I thought that groups with the same Lie Algebra are automatically equivalent, but there appear to be some exceptions to this?
What sort of exceptions are there and why?
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What definition of "equivalent" are you thinking of? – Santiago Canez Oct 16 '15 at 16:34
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Have the same group structure. Are the same group. – MadScientist Oct 16 '15 at 16:37
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3The additive group of real numbers and the multiplicative group of complex numbers of absolute value have isomorphic Lie algebras. In general, Lie groups with isomorphic Lie algebras are locally isomorphic, but they need not be globally isomorphic. – Andreas Blass Oct 16 '15 at 16:38
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What are the there situations where they are not locally isomorphic? – MadScientist Oct 16 '15 at 16:45
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1You need to assume your Lie groups are simply connected. Then you are correct. – Espen Nielsen Oct 16 '15 at 16:47
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The 3-sphere (unit quaternions) and SO(3) have the same Lie algebra. Why? Because one is a double cover of the other, and the Lie algebra can be defined purely locally (via vector fields, for instance).
So I guess a general answer is "covering groups are a general case where same algebra doesn't (necessarily) mean same group."
(Sometimes you get a covering group that IS isomorphic, as in the covering $\theta \mapsto 2\theta$ on $S^1$.)
John Hughes
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Do you believe there are other answers or does this answer exhaust all possibilities? – MadScientist Oct 16 '15 at 17:00
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See Espen Nielsen's comment; that suggests that this is the only possibility. – John Hughes Oct 16 '15 at 18:34
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6Each (finite-dimensional) Lie algebra has exactly one simply connected Lie group associated to it (up to isomorphism). All other connected groups with the same Lie algebra are quotients of the simply connected one by discrete central subgroups. (See my Introduction to Smooth Manifolds, 2nd ed., Theorem 21.32 and Problem 21.18.) – Jack Lee Oct 16 '15 at 21:53