Timeline for Several question on Affine Lie algebra
Current License: CC BY-SA 2.5
10 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Apr 23, 2010 at 12:20 | comment | added | Jim Humphreys | @Shizhuo: Sources for self-study of Kac-Moody theory are somewhat limited, but besides the book by Kac there are useful books by Moody-Pianzola and more recently Roger Carter (Cambridge Press). Carter's book is quite readable, but is limited to the most applicable cases: finite dimensional and affine Lie algebras. | |
| Apr 23, 2010 at 12:16 | comment | added | Jim Humphreys | @Ben: My first comment was indeed misguided; I was thinking of a more complicated problem. The highest weights behave well enough, though the analysis of infinitely many "composition factors"is problematic. | |
| Apr 23, 2010 at 4:31 | vote | accept | Shizhuo Zhang | ||
| Apr 23, 2010 at 4:05 | answer | added | S. Carnahan♦ | timeline score: 5 | |
| Apr 23, 2010 at 2:17 | comment | added | Ben Webster♦ | @Jim- What's wrong with the tensor product in the usual case? It's still an integrable module with finite dimensional weight spaces (of course, it can have infinitely many isotypic components, which is a bit weird). | |
| Apr 22, 2010 at 17:47 | comment | added | Shizhuo Zhang | @Jim, thank you for your caution on tensor products of infinite dimensional space. | |
| Apr 22, 2010 at 17:35 | comment | added | Jim Humphreys | 1) More care is needed about the meaning of tensor product of two irreducible representations in the Kac-Moody case, even when the modules involved are "integrable" and exhibit some of the classical character behavior. Tensoring infinite dimensional modules is quite a subtle question. (I'm not at all a specialist, however.) 2) As Emerton points out, you don't need Weyl's character formula in the classical case, but of course you do need well-behaved tensor products. | |
| Apr 22, 2010 at 16:00 | comment | added | Emerton | Perhaps you know this, but to show that $R(\lambda)\otimes R(\mu)$ maps to $R(\lambda + \mu)$, you don't need anything as sophisticated as a character formula. Here is a more robust argument, which might extend to other contexts: Simply note that if $e_{\lambda}$ and $e_{\mu}$ are the highest weight vectors, then $e_{\lambda} \otimes e_{\mu}$ is a highest weight vector in the tensor product, which must then generate a copy of $R(\lambda + \mu)$ inside the tensor product. The existence of a projection now follows from semi-simplicity of the representation category. | |
| Apr 22, 2010 at 14:54 | history | asked | Shizhuo Zhang | CC BY-SA 2.5 |