Timeline for Can we prove that there are countably many isomorphism classes of compact Lie groups without the classification of simple Lie algebras?
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Jul 20, 2013 at 5:11 | answer | added | Nicola Ciccoli | timeline score: 2 | |
| Jul 4, 2013 at 0:13 | comment | added | Qiaochu Yuan | @Jim: yep. ${}{}$ | |
| Jul 3, 2013 at 23:39 | comment | added | Jim Humphreys | @Qiaochu: Note that it's a standard fact that semisimple complex Lie algebras are "rigid" among all Lie algebras of a given dimension. Is this equivalent to what you are observing in the compact case? | |
| Jul 2, 2013 at 18:13 | comment | added | Qiaochu Yuan | @Jim: not really, but I expect to learn something interesting from the techniques used in an answer. For example, an easier result I suspect I already have the tools to prove is that the Lie algebras of compact Lie groups are infinitesimally rigid in the sense that they have no first-order deformations (I think this follows using a computation in Lie algebra cohomology). | |
| Jul 2, 2013 at 15:49 | comment | added | Jim Humphreys | @Qiaochu: This kind of question is fun to think about and certainly not straightforward to answer, but would an answer one way or the other have any interesting consequences? (It's a bit like noting that there can be only countably many isomorphism classes of finite simple groups. How far does that get one?) | |
| Jun 29, 2013 at 7:46 | comment | added | Qiaochu Yuan | @Mariano: also, do you know if the results Claudio cites are independent of the classification? They sound like the kind of thing you could prove just by casework using the classification. | |
| Jun 29, 2013 at 7:44 | comment | added | Mariano Suárez-Álvarez | @QiaochuYuan, the last step should be true, but I couldn't find a reference stating it :-/ Maybe some of the expects around can fill in that hole. | |
| Jun 29, 2013 at 7:40 | comment | added | Qiaochu Yuan | Real compact Lie group and isomorphism of real Lie groups respectively. @Mariano: great! Do you want to post that as an answer or do you have some reservations about the last step? | |
| Jun 29, 2013 at 4:14 | comment | added | Pete L. Clark | Note that this seems to depend upon what you mean by "compact Lie group" and "isomorphism", since complex tori vary in moduli. | |
| Jun 28, 2013 at 21:06 | comment | added | Mariano Suárez-Álvarez | Since compact Lie groups have finite triangulations, there are countably many candidates for the underlying topological space up to homeomorphism. It follows from the result quoted at the end of Claudio's answer here that there are then countably many options for the Lie algebra. That gives, by Lie theory, countably many candidates for the isomorphism type of the universal covers of compact Lie groups, and each of them has at most countably many compact quotients with discrete kernel (I think). | |
| Jun 28, 2013 at 20:58 | history | asked | Qiaochu Yuan | CC BY-SA 3.0 |