Timeline for Better names for Lie groups
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Jan 31, 2019 at 6:49 | comment | added | David Roberts♦ | @Marek no worries :-) and thanks for pointing out an older reference, I didn't bother chasing down original authors. | |
| Jan 31, 2019 at 6:29 | comment | added | user21230 | @DavidRoberts Sorry, I criticise your comments, not you (talk to Asaf). I think already Cartan or Dickson has defined $E_7$ as group preserving some 4-linear form on 56-dimensional complex space. See e.g. Dickson, "The configuration of the 27 lines...", 1901 | |
| Jan 30, 2019 at 7:17 | comment | added | user21230 | @timothychow "No" is valid answer. However we can ask a thousand next questions. What have mathematicians been trying instead ? Are they happy with current definitions of exceptional Lie groups ? | |
| Jan 30, 2019 at 7:15 | comment | added | David Roberts♦ | @Marek beats me! I just happened to have read a review article in the Bulletin AMS recently that mentioned these results, so I thought I'd point them out. | |
| Jan 30, 2019 at 7:13 | comment | added | user21230 | @davidroberts I read some Freudenthal papers. It is not easy to follow. I believe, he constructed not compact version of the group - at least in "Beziehungen...". I am looking for more direct definition of the group as matrix group over quaternions. What geometry can be found behind the quartic form which you mention ? | |
| Jan 30, 2019 at 3:22 | comment | added | Timothy Chow | I like this question but I agree with Anton Fetisov that it does not seem to be focused enough for MathOverflow. Probably the answer to the question as stated ("have mathematicians tried to invent...") is a simple "no." | |
| Jan 29, 2019 at 23:41 | comment | added | David Roberts♦ | For exceptional Lie groups it is not easy neither to define the group nor the structure. <--- See eg Theorem 8.1 in GARIBALDI, S., & GURALNICK, R. (2015). SIMPLE GROUPS STABILIZING POLYNOMIALS. Forum of Mathematics, Pi, 3, E3. doi.org/10.1017/fmp.2015.3 || For $E_7$, Freudenthal's paper ‘Sur le groupe exceptionnel $E_7$’, Nederl. Akad. Wetensch. Proc. Ser. A 56 = Indagationes Math. 15 (1953), 81–89. gives a quartic polynomial on the vector space underlying a the smallest nontrivial rep of $E_7$ (not clear whether complex or compact) whose automorphisms give the group. | |
| Jan 29, 2019 at 12:56 | comment | added | Ben McKay | @AntonFetisov: the higher projective spaces are defined as homogenous spaces. The octonionic projective plane is a projective plane according to Hilbert's definition (en.wikipedia.org/wiki/Projective_plane), but the others are not, sadly. | |
| S Jan 29, 2019 at 12:45 | history | suggested | LeechLattice |
add soft-question tag
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| Jan 29, 2019 at 12:20 | review | Suggested edits | |||
| S Jan 29, 2019 at 12:45 | |||||
| Jan 29, 2019 at 12:04 | history | edited | user21230 | CC BY-SA 4.0 |
added 1132 characters in body
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| Jan 29, 2019 at 11:58 | history | edited | user21230 | CC BY-SA 4.0 |
added 1132 characters in body
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| Jan 29, 2019 at 11:31 | comment | added | Anton Fetisov | This seems an invitation to discussion rather than a focused question, so it appears to be offtopic on this site. However, note that the projective spaces in your question aren't really well-defined (neither as projectivizations of some space nor as some algebraic spaces), but rather appear in a very ad-hoc fashion as some special Jordan algebras. Nor are the higher projective spaces over bioctonions etc defined, really. This means there is still no natural series here but rather very different and disconnected examples with only an intuition connecting them. | |
| Jan 29, 2019 at 10:55 | history | asked | user21230 | CC BY-SA 4.0 |