Timeline for Counting degrees of freedom in Lie algebra structure constants (aka why are there any nontrivial Lie algebras of dim >5?)
Current License: CC BY-SA 4.0
11 events
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| Jan 20, 2023 at 15:04 | comment | added | Kensmosis | @TimothyChow Thanks for the suggestion. I am indeed new to varieties. I'd heard of them, but never realized they were this interesting or useful. Coincidentally, I just encountered the twisted cubic at the beginning of Harris' intro book on the subject. It's definitely clear I was being naive in my approach to the Jacobi eqs, and now I'm trying to get a basic flavor for how one does go about tackling such problems in general. | |
| Jan 19, 2023 at 13:45 | comment | added | Timothy Chow | @Kensmosis If you're new to the concept of a variety that is not a complete intersection, the textbook example is the twisted cubic. It's an illuminating exercise to try (unsuccessfully, of course) to express it using only two equations, and to explain to yourself why having three equations doesn't reduce it to a zero-dimensional object. | |
| Jan 19, 2023 at 3:22 | vote | accept | Kensmosis | ||
| Jan 19, 2023 at 3:21 | comment | added | Kensmosis | The Kirillov paper does give me some sense of the issue involved (and Qiaochu Yuan's example below also helped me grasp it more clearly). It's clear I need to learn a bit about algebraic varieties before being able to think sensibly about the Jacobi equations. My thanks to you (and everyone else who chimed in) for pointing me in the right direction! | |
| Jan 19, 2023 at 3:11 | comment | added | Kensmosis | Thanks for the additional references! A proof of linearity is much more satisfying than some quick and dirty computer program, and I'm looking forward to checking it out. | |
| Jan 18, 2023 at 13:26 | comment | added | Timothy Chow | The Kirillov-Neretin paper notes that already in dimension 3, there exists a component that is not a complete intersection. This fact partially addresses your question of why we can't simply compute the dimension by subtracting the number of equations from the number of variables. | |
| Jan 18, 2023 at 8:52 | comment | added | Vladimir Dotsenko | @Kensmosis I added two more references that you will perhaps find useful | |
| Jan 18, 2023 at 8:51 | history | edited | Vladimir Dotsenko | CC BY-SA 4.0 |
added two references
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| Jan 17, 2023 at 21:10 | comment | added | Kensmosis | I agree that linear independence says very little, but I figured I should at least do that basic level of due-diligence before bothering anyone with my Q. Thanks for the references. I haven't studied algebraic varieties but will take a look. Kirillov's book is where I learned Lie Groups and Lie Algebras, so hopefully his paper will prove somewhat accessible to me. | |
| Jan 17, 2023 at 18:43 | history | edited | LSpice | CC BY-SA 4.0 |
Names of papers
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| Jan 17, 2023 at 18:16 | history | answered | Vladimir Dotsenko | CC BY-SA 4.0 |