Timeline for A possible mistake in Kac's "Infinite Dimensional Lie Algebras"
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Jul 29, 2012 at 15:25 | answer | added | Jim Humphreys | timeline score: 11 | |
| Jul 27, 2012 at 17:52 | comment | added | Jim Humphreys | @Najdorf: [Note I called C the "Tits cone", but that label belongs to the W -saturated set X. Definite "mistake".] Since X' is closed and is shown to contain X, it also contains the closure of X. The proof isn't written down carefully, but I think if you assemble the various steps they do add up to a correct proof of the reverse inclusion for the closure of X. ["Misprint"?] This book is known for its rough spots, even the 3rd edition, but it's an essential source. The only similar textbooks are those by Moody-Pianzola and Carter, which have different emphases. Good luck | |
| Jul 27, 2012 at 13:45 | comment | added | Najdorf | I used the word "mistake" because I think the statement is false and you should replace the closure with the cone. Of course I might be mistaken myself, hence "possible mistake". | |
| Jul 27, 2012 at 13:43 | comment | added | Najdorf | But if you read the proof he says let X^{\prime} be the right hand side and then shows that it is equal to X itself! The closure in metric topology is not mentioned even once! | |
| Jul 27, 2012 at 13:34 | comment | added | Jim Humphreys | I glanced at that passage in the 3rd edition, which gets convoluted by the time you reach part (c) treating the case of an indefinite matrix A. (The finite and affine cases, which come up more often in applications, are a little more straightforward.) Keep in mind that the Tits cone C is defined in general by non-strict inequalities (so it serves as a fundamental domain). But the presence of imaginary roots complicates life, so you have to refer back to earlier sections and assemble steps in (c) very carefully. Probably "mistake" is too strong, but it's not easy to follow. | |
| Jul 27, 2012 at 6:18 | history | asked | Najdorf | CC BY-SA 3.0 |