Timeline for Do compact inverse-property loops (or just compact Moufang loops) have invariant uniformities and bi-invariant Haar measure?
Current License: CC BY-SA 4.0
21 events
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| Apr 18 at 1:53 | history | edited | Harry Altman | CC BY-SA 4.0 |
apparently this use of "left-invariant" is actually standard, so I'm updating the question (and title) to reflect that
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| Apr 5 at 7:36 | comment | added | Harry Altman | Worth noting: There are, apparently, compact loops without invariant uniformities (and I could probably adapt the examples to prove a lack of Haar measure, too). But the only examples I've seen don't have the inverse property, so... | |
| Jul 10, 2023 at 6:13 | comment | added | მამუკა ჯიბლაძე | @TarasBanakh Spectacular! Please if you recall a reference, let me know! On Wikipedia I only found that the group of self-isotopies is Spin(8), and that of unit-preserving isotopies is Spin(7). Maybe this already suffices but what you say is still very interesting | |
| Jul 10, 2023 at 5:50 | comment | added | Taras Banakh | @მამუკაჯიბლაძე As far as I remember they are related through a suitable section $s:G/H\to G$ of the quotient map $q:G\to G/H$. So that $x*y=q(s(x)\cdot s(y))$ where $*$ is the loop multiplication and $\cdot $ is the multiplication in the group. | |
| Jul 9, 2023 at 22:14 | comment | added | მამუკა ჯიბლაძე | @TarasBanakh What I don't see is how the homogeneous space structure relates to the loop multiplication - a priori they are totally unrelated, no? | |
| Jul 8, 2023 at 18:30 | comment | added | Taras Banakh | @მამუკაჯიბლაძე I hope so. The invariance (at least one-sided) of the quotient measure should follow from the invariance of the Haar measure. | |
| Jul 8, 2023 at 7:01 | comment | added | მამუკა ჯიბლაძე | @TarasBanakh Also it is not clear (to me) what kind of representation of, say, the loop of unit octonions as a quotient of some topological group would give an invariant measure. It is a 7-dimensional sphere, so it is, for example, $SO(8)/SO(7)$, but is the resulting measure invariant? | |
| Jul 8, 2023 at 2:34 | history | edited | Harry Altman | CC BY-SA 4.0 |
add missing words
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| S Jul 7, 2023 at 9:03 | history | bounty ended | CommunityBot | ||
| S Jul 7, 2023 at 9:03 | history | notice removed | CommunityBot | ||
| S Jun 29, 2023 at 7:28 | history | bounty started | Harry Altman | ||
| S Jun 29, 2023 at 7:28 | history | notice added | Harry Altman | Draw attention | |
| S Dec 10, 2021 at 9:03 | history | bounty ended | CommunityBot | ||
| S Dec 10, 2021 at 9:03 | history | notice removed | CommunityBot | ||
| Dec 7, 2021 at 17:44 | comment | added | Taras Banakh | Unfortunately, I have not elaborated my coment to the level sufficient for a good answer. Now I am even not sure about the representation of a zero-dimensional loop as an inverse limit of finite loops. This is a good question if such a representation exists for every compact zero-dimensional loop. For a bit different structure, namely that of a dynamical system, such representations do not exist: just look at the Cantor cube $\{0,1\}^Z$ with the shift map. | |
| Dec 7, 2021 at 7:07 | comment | added | Harry Altman | Hm, that's certainly a quite different approach. I can try it, but any chance you can fill in the details enough to turn that into an answer...? | |
| Dec 2, 2021 at 8:43 | comment | added | Taras Banakh | For the general case, one should represent a compact loop and the quotient space of some topological group (generated by shifts) and look for such an invariant measure on the homogeneous space. | |
| Dec 2, 2021 at 8:42 | comment | added | Taras Banakh | For finite loops you have such a measure. Since each zero-dimensional compact loop is an inverse limit of finite loops, such an invariant measure should exist also for compact zero-dimensional loops. | |
| S Dec 2, 2021 at 7:14 | history | bounty started | Harry Altman | ||
| S Dec 2, 2021 at 7:14 | history | notice added | Harry Altman | Draw attention | |
| Nov 30, 2021 at 5:35 | history | asked | Harry Altman | CC BY-SA 4.0 |