Timeline for Discrete-compact duality for nonabelian groups
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| Jul 22, 2011 at 21:05 | answer | added | Andreas Thom | timeline score: 4 | |
| Jul 21, 2011 at 21:49 | comment | added | Qiaochu Yuan | @Yemon: you can always take orthogonal complements in a unitary representation. What you lose in the general case, of course, is the guarantee that unitary representations separate points... | |
| Jul 21, 2011 at 21:34 | comment | added | Yemon Choi | Theo: is it really true that sub-objects of unitary representations are direct summands for all groups? I would have expected such a result to require some kind of compactness or amenability, but perhaps I am missing something obvious | |
| Jul 19, 2011 at 12:17 | answer | added | Marc Palm | timeline score: 5 | |
| Jul 19, 2011 at 7:38 | comment | added | David Roberts♦ | If you are looking in the direction of generalised spaces with a notion of compactness and discreteness, try locales. :) | |
| Jul 19, 2011 at 6:31 | answer | added | Anatoly Kochubei | timeline score: 10 | |
| Jul 19, 2011 at 2:33 | comment | added | Theo Johnson-Freyd | Note: it is probably convenient to work only with unitary representations, rather than all of them, because the category of unitary representations of any group is completely reducible (sub-objects are direct summands), whereas non-compact groups also have non-semisimple representation theory. The upside of complete reducibility is that the category is still controlled by the space of irreducible representations. Note that it is essentially a space, and not some higher stack, because Schur's lemma assures that the points do not have more automorphisms than you would expect. | |
| Jul 19, 2011 at 2:28 | comment | added | Theo Johnson-Freyd | Great question! I expect that the answer is something close to what you're looking for. It is true that (1) every representation of a compact group is unitarizable, by averaging, and (2) the irreducible representation theory is discrete. For noncompact groups, a good warm-up is to consider the case of SL(2,R). Then (1) there are non-unitarizable representations (all nontrivial finite-dimensional ones), and (2) there are continuous families of irreducible unitary representations (the infinite-dimensional representations). | |
| Jul 19, 2011 at 1:26 | answer | added | Justin Campbell | timeline score: 1 | |
| Jul 19, 2011 at 0:26 | comment | added | Theo Buehler | This seems closely related to Andreas's answer, but I never found a reference putting the finger onto a precise relation. There's the book Kac algebras and duality of locally compact groups by Enock-Schwartz, where a Pontryagin-style duality is developed for Kac-Algebras (essentially Hopf algebras which are at the same time von Neumann algebras): ams.org/mathscinet-getitem?mr=1215933 | |
| Jul 19, 2011 at 0:14 | answer | added | anon | timeline score: 0 | |
| Jul 18, 2011 at 23:21 | comment | added | paul garrett | I like the formulation with a one-parameter family of unitaries... | |
| Jul 18, 2011 at 22:54 | comment | added | KConrad | I really doubt that spelling with j instead of y is going to make the name in English pronounced more accurately by most English speakers. In any case, languages aren't required to make foreign names or places sound like they do in the native language. It'd be nicer if they are, but if not, well, c'est la vie. | |
| Jul 18, 2011 at 22:07 | comment | added | Qiaochu Yuan | Both spellings seem to be pretty widely used. My impression is that Pontrjagin is a slightly more faithful romanization. "Pontryagin" invites a pronounciation in which the "y" and the "a" are pronounced separately, but in Cyrillic "ja" is one character and, as I understand it, one syllable. | |
| Jul 18, 2011 at 21:39 | comment | added | user16553 | Qiaochu the spelling should be Pontryagin Duality. | |
| Jul 18, 2011 at 21:05 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Jul 18, 2011 at 20:54 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Jul 18, 2011 at 19:02 | answer | added | Andreas Thom | timeline score: 16 | |
| Jul 18, 2011 at 18:48 | history | asked | Qiaochu Yuan | CC BY-SA 3.0 |