Timeline for Universal cover of SL2(R) admits no central extensions?
Current License: CC BY-SA 4.0
10 events
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| Aug 24, 2022 at 11:46 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
a minor typo
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| May 1, 2011 at 14:40 | comment | added | Benjamin Westrich | Ok, I think the answer of Richard Borcherds already clarified the issue. I supected the proof could be found in Steinbergs original papers but I didn't find them when I was searching... | |
| Apr 30, 2011 at 18:07 | comment | added | Junkie | Here is a nice reference: users.ictp.it/~pub_off/lectures/lns023/Rehmann/Rehmann.pdf It notes (Remark 1, page 89) that the Steinberg group is perfect, so maybe my above offering is not too far out. | |
| Apr 30, 2011 at 18:04 | comment | added | Junkie | I think the answer is yes. At the group theory level, a central extension $E$ is a cover if $[E,E]=E$, that is $E$ is perfect. So you can take the Steinberg group for $SL_2(R)$, which is known as universal central, and show directly(?) that it is perfect, thus it is also a cover, contained universally. Just an idea. I found nothing but vague unreferenced material like math.psu.edu/katok_a/pub/CR-publ.pdf "In general, for a connected real analytic semisimple Lie group the usual Lie group theoretic simply connected cover is its universal topological central extension." | |
| Apr 30, 2011 at 17:33 | comment | added | André Henriques | I suspect that, in all those references, the words "universal central extension" mean "universal central extension in the category of Lie groups", and not "universal central extension in the category of groups". | |
| Apr 30, 2011 at 16:14 | comment | added | Benjamin Westrich | I guess they are not happy with that. I will look for an appropriate reference. | |
| Apr 30, 2011 at 16:10 | comment | added | Junkie | Not happy with Wikipedia, with no other source given? :) "The Schur multiplier of PSL(2,R) is Z, and the universal central extension is the same as the universal covering group." en.wikipedia.org/wiki/SL2%28R%29 | |
| Apr 30, 2011 at 16:02 | comment | added | André Henriques | If what you say is correct, then that answers my question. But I would like to see an argument (or a reference) for the fact that the universal cover for SL2(R) is also its universal central extension (as an abstract group). | |
| Apr 30, 2011 at 15:49 | comment | added | Alain Valette | Can you provide a reference? Thanks. | |
| Apr 30, 2011 at 15:20 | history | answered | Benjamin Westrich | CC BY-SA 3.0 |