Timeline for Motivating the Casimir element
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 30, 2020 at 8:46 | comment | added | jjcale | In quantum mechanics the casimir element is very natural since in the case of the Lie groups SO(3) und SU(2) it corresponds to the squared total angular momentum . | |
| May 29, 2020 at 23:13 | answer | added | Ishan Levy | timeline score: 11 | |
| Dec 13, 2013 at 7:47 | answer | added | Daniel Litt | timeline score: 37 | |
| Sep 8, 2011 at 14:19 | vote | accept | Timothy Chow | ||
| Sep 7, 2011 at 22:46 | answer | added | Jim Humphreys | timeline score: 85 | |
| Sep 7, 2011 at 1:33 | answer | added | Moosbrugger | timeline score: 58 | |
| Sep 7, 2011 at 0:33 | answer | added | Ryan Reich | timeline score: 14 | |
| Sep 6, 2011 at 22:39 | comment | added | Ryan Reich | Perhaps not, but I meant that in a structured development of the subject, you'd have "Chapter: the universal enveloping algebra" somewhere, and that would have the PBW theorem in it. A more conversational book could break it up. In a structured development, you would also have a pre-book presenting the theory of modules over an arbitrary noncommutative unital ring, too, the lack of which is probably another reason that the representation theory of $\mathfrak{g}$ is not often built on that of $U(\mathfrak{g})$. This may be a backlask against Bourbakism after all, but not considered as such. | |
| Sep 6, 2011 at 22:23 | comment | added | darij grinberg | Is PBW really necessary to work with the Casimir here? | |
| Sep 6, 2011 at 22:18 | comment | added | Ryan Reich | @darij: More likely, it's for the reason that Humphreys himself gives later on when he does get to $U(\mathfrak{g})$: because the technical overhead of proving the PBW theorem makes it not worth introducing the subject before it's absolutely necessary. Of course, Humphreys' book is introductory so this makes some kind of sense; on a theoretical level I don't know why you wouldn't slap down $U(\mathfrak{g})$ in chapter 1. | |
| Sep 6, 2011 at 22:00 | comment | added | darij grinberg | Note that I am talking about the definition of Casimir element given here: amathew.wordpress.com/2010/01/31/… . Humphreys obscures this somewhat by avoiding the use of $U\left(\mathfrak g\right)$; I personally find this unfortunate, but it seems to be a popular things to do (a backlash to Bourbakism?). | |
| Sep 6, 2011 at 21:57 | comment | added | darij grinberg | Interesting... I thought the Casimir element was the most natural part of the proof! We have the semisimplicity, which says that the Killing form is nondegenerate. We have to use it somewhere, but how can we relate it to a representation? Well, we can encode it into an element of $\mathfrak g\otimes \mathfrak g$, and map this canonically into $U\left(\mathfrak g\right)$ by the map $x\otimes y\mapsto xy$. The resulting element of $U\left(\mathfrak g\right)$ then acts on representations of $\mathfrak g$. | |
| Sep 6, 2011 at 21:47 | history | asked | Timothy Chow | CC BY-SA 3.0 |