Timeline for Realizing root-system roots as polynomial roots without Lie theory
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Mar 14, 2018 at 14:40 | comment | added | Jim Humphreys | @Mike: I should clarify that it's probably noit realistic to discuss root systems abstractly without some mention of Lie algebras. The notion of "root" does arise from the characteristic polynomials of adjoint operators and their roots. Without this motivation, why would anyone study "root systems" axiomatically? (On te other hand, the abstract definition allows one to generalize to infinite-dimensional Lie algebras of Kac-Moody type, etc.) | |
| Mar 6, 2018 at 1:44 | history | edited | Mike Pierce | CC BY-SA 3.0 |
Added the a sentence
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| Mar 4, 2018 at 2:23 | history | edited | Mike Pierce | CC BY-SA 3.0 |
Added some motivation
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| Mar 4, 2018 at 1:42 | comment | added | Mike Pierce | @JimHumphreys I figured that this question either had a nice clean answer that I wasn't seeing, or really no answer at all. Based on what you're saying, it seems like the latter though, so there isn't much to learn. The motivation for this question was to have a response to the question "why are they called roots?" when talking about abstract root systems with students who might not have a background in Lie theory. | |
| Mar 4, 2018 at 0:28 | comment | added | Jim Humphreys | I'm not clear about the motivation for going in this direction, but it's probably feasible at least case-by-case if one first classifies the simple Lie algebras and then contructs them individually in known ways. Anyway, the Bourbaki/Serre notion of abstract root system grew out of early work by Killing and Cartan, with intermediate steps by Jacobson and others. But what is learned by going in the reverse direction? | |
| Mar 3, 2018 at 20:57 | answer | added | Ben Webster♦ | timeline score: 1 | |
| Mar 3, 2018 at 0:55 | comment | added | Will Jagy | Alright. He's on this site; it appears he is active from time to time. Maybe he will notice this. | |
| Mar 3, 2018 at 0:38 | history | edited | Mike Pierce | CC BY-SA 3.0 |
Added (crystallographic)
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| Mar 3, 2018 at 0:37 | comment | added | Mike Pierce | @WillJagy Yeah, that's where I started. Humphreys mentions that the term "root" historically comes from Lie theory (section 1.2), and then makes the distinction between general root systems and crystallographic roots systems that relate to Lie theory (section 2.9). I haven't seen the answer to the question spelled out anywhere though. | |
| Mar 2, 2018 at 23:17 | comment | added | Will Jagy | I can recommend Reflection Groups and Coxeter Groups by James Humphreys. I am flipping through it and cannot guarantee that it answers your concerns. On the other hand, I know of no book called Root Systems and Polynomials, so Humphreys is a good start. Plenty of roots, plenty of polynomials. | |
| Mar 2, 2018 at 20:07 | history | edited | Mike Pierce | CC BY-SA 3.0 |
added 58 characters in body
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| Mar 2, 2018 at 19:52 | history | asked | Mike Pierce | CC BY-SA 3.0 |