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
10 events
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| Jan 22, 2023 at 8:27 | comment | added | Qiaochu Yuan | @printf: in that example the conditions you're imposing are all linear so you only need to check that they're linearly independent. The point of this discussion is that for nonlinear constraints the relevant notion of "independence" is considerably more subtle and hard to check. | |
| Jan 22, 2023 at 6:34 | comment | added | user496902 | I have to say, now I am slightly concerned... in physics (where I come from) we often count the number of (linearly independent) tensors of given type using similar arguments. For example, to count the number of completely antisymmetric and traceless tensors (antisymmetric under exchange of any pair of indices, traceless for any pair of indices) we would count the number of antisymmetric tensors, then subtract the number of "tracelessness conditions" that we need to impose. Does this mean that this procedure may not always be justified? | |
| Jan 20, 2023 at 16:49 | comment | added | Vladimir Dotsenko | @Kensmosis no, intersection theory is much deeper than things used in discussions here. A good headline to begin with is dimension theory. In particular, "complete intersection" is an intersection where adding each of the equations reduces the dimension by one, and to understand it well, it is of utmost importance to have full clarity on different ways of how one can think of dimension in algebraic geometry. | |
| Jan 20, 2023 at 14:55 | comment | added | Kensmosis | @VladimirDotsenko Thanks for the distilled example! I've begun reading a simple intro to the subject (Harris' Invitation to Algebraic Geometry). If you have suggestions for a more directly-relevant (but still introductory) treatment of the specific topics I'd need to tackle these sorts of problems, I'd be very interested. From what I gather, it's called intersection theory --- but that seems to be a fairly broad area of algebraic geometry. | |
| Jan 19, 2023 at 19:30 | history | edited | Qiaochu Yuan | CC BY-SA 4.0 |
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| Jan 19, 2023 at 10:37 | comment | added | Vladimir Dotsenko | @Kensmosis For pedagogical purposes, it is also useful to consider the example $x_ix_j=0$ with $(i,j)\ne (1,1)$. There are $n$ unknowns and $\binom{n+1}{2}-1$ linearly independent equations, and still there is a nontrivial solution $(1,0,\ldots,0)$. | |
| Jan 19, 2023 at 3:13 | comment | added | Kensmosis | Thanks, that's a very illuminating example (and way easier for me to visualize than the Lie Algebra case). Sounds like I need to study algebraic varieties a bit to get a better handle on what's happening in the Lie Algebra case. | |
| Jan 18, 2023 at 23:24 | history | edited | Qiaochu Yuan | CC BY-SA 4.0 |
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| Jan 18, 2023 at 23:14 | history | edited | Qiaochu Yuan | CC BY-SA 4.0 |
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| Jan 18, 2023 at 23:06 | history | answered | Qiaochu Yuan | CC BY-SA 4.0 |