Timeline for Motivation for Hall-Witt identity
Current License: CC BY-SA 3.0
13 events
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| May 20, 2023 at 23:03 | comment | added | DavidLHarden | mathshistory.st-andrews.ac.uk/Groups/2017/slides/cohen-b.pdf has a 4-variable version of Hall-Witt, but it's cyclically symmetric. For aesthetic reasons, I'd prefer a 4-variable version where the 4 factors are images of one another under the action of V. | |
| Mar 17, 2018 at 10:54 | history | edited | YCor |
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| Mar 17, 2018 at 10:30 | answer | added | Tim Porter | timeline score: 4 | |
| Jun 6, 2017 at 11:32 | comment | added | Glasby | math.stackexchange.com/questions/251955/… | |
| Jan 25, 2012 at 22:04 | comment | added | Marty Isaacs | A two variable version of Hall-Witt is the nearly trivial identity [a,b][b,a] = 1. Hall-Witt, of course, has three variables. This suggests the question of the existence of some four variable version. I wonder if some kind of computer search might yield an identity of this kind. | |
| Nov 20, 2011 at 17:37 | comment | added | Terry Tao | lamington.wordpress.com/2011/11/20/the-hall-witt-identity | |
| Nov 18, 2011 at 16:54 | comment | added | Colin Reid | When it comes to such identities in general, what you are really trying to understand is the structure of free groups. The theory of free groups has a strong geometric flavour, so perhaps there is a geometric interpretation of the Hall-Witt identity that makes it seem more natural. | |
| Nov 18, 2011 at 9:48 | comment | added | Alireza Abdollahi | @Martin $[a,b,c]=[[a,b],c]$, where $[a,b]=a^{-1}b^{-1}ab$. | |
| Nov 18, 2011 at 8:46 | comment | added | André Henriques | Well... the motivation for introducing it might be that 1) it looks like the Jacobi identity 2) from it, you can derive the Jacobi identity for the Lie algebra of an algebraic group 3) from it, you can derive the Jacobi identity for the Lie algebra associated to the lower central series of a discrete group. | |
| Nov 18, 2011 at 8:32 | comment | added | Martin Brandenburg | @Alireza: What is $[a,b,c]$? | |
| Nov 18, 2011 at 8:31 | history | edited | Martin Brandenburg |
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| Nov 18, 2011 at 7:23 | comment | added | Alireza Abdollahi | It may be intersting to note that if $G$ is metaablian (i.e. the derived subgroup $[G,G]$ is abelian), then the Hall-Witt identity looks like exactly as the Jacobi identity, that is to say you may ignore and drop the conjugates and the inverses in the identity; That is $$[a,b,c][b,c,a][c,a,b]=1,$$ holds for all $a,b,c\in G$ whenever $G$ is metaabelian. | |
| Nov 18, 2011 at 6:53 | history | asked | Selim | CC BY-SA 3.0 |