Timeline for The normalised form for the twisted Kac-Moody algebra
Current License: CC BY-SA 3.0
12 events
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| May 25, 2017 at 13:59 | comment | added | Jim Humphreys | @Paul Levy: Apparently there are a lot of differences among the editions of the Kac book; I have only the 3rd edition, published by Cambridge in 1990. | |
| May 25, 2017 at 13:56 | comment | added | Z.A.Z.Z | Just before Lemma 8.5, he uses the identification $$Aut(\mathcal L(\mathfrak{sl}_r))\cong Mor(\mathbb C^\times \rightarrow Aut(\mathfrak{sl}_r)).$$ I expect your version is quite different. | |
| May 25, 2017 at 13:43 | comment | added | Paul Levy | This doesn't seem to me to be a statement which is equivalent to Remark 8.5. I have the 2nd edition of Kac's book and as far as I can see (without reading it in detail) it doesn't say anything about automorphisms of the loop algebra, only about automorphisms of the finite-dimensional Lie algebra. | |
| May 25, 2017 at 13:26 | comment | added | Z.A.Z.Z | But, as I mentioned in the last comment above, by Lemma 8.5, $\tau$ and $\sigma$ give isomorphic twisted Kac-Moody algebras iff $$\tau=f(t)\sigma f(-t)^{-1},$$ for some $f(t)\in Aut(\mathcal L(\mathfrak{sl}_r))$, this is not clear for me, I don't see such $f(t)$. | |
| May 25, 2017 at 13:02 | comment | added | Paul Levy | Right, well there you go - these are both outer automorphisms so they will give isomorphic twisted loop algebras. (There are only two connected components of ${\rm Aut}(\mathfrak{sl}_n)$.) | |
| May 25, 2017 at 12:57 | comment | added | Z.A.Z.Z | Remark 8.5: sais and I quote: "... the isomorphism class of $\mathcal L(\mathfrak{g},\sigma,m)$ depends only on the connected component of $Aut(\mathfrak{g})$ containing $\sigma$. .... " which is a consequence of Proposition 8.5 (if obviously it is the same as that in your version). By the way I have the 3rd version. | |
| May 25, 2017 at 12:30 | comment | added | Paul Levy | Whoops - that got left in there from an earlier edit. There was also a mistake in that sentence, which I think I have fixed now. | |
| May 25, 2017 at 12:29 | history | edited | Paul Levy | CC BY-SA 3.0 |
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| May 25, 2017 at 12:21 | comment | added | Z.A.Z.Z | Thank you for your answer. It is just a metter of notation, I usually denote the affine Lie Algebra by $\widehat{\mathfrak{sl}}_r(K)$ where $K=\mathbb C((t))$. I assume that $e_{\alpha_i}$ are the Chevalley generators... also what is $\gamma$? | |
| May 25, 2017 at 12:14 | history | edited | Paul Levy | CC BY-SA 3.0 |
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| May 25, 2017 at 10:23 | history | edited | Paul Levy | CC BY-SA 3.0 |
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| May 25, 2017 at 10:18 | history | answered | Paul Levy | CC BY-SA 3.0 |