Timeline for Explicit cocycle for the central extension of the algebraic loop group G(C((t)))
Current License: CC BY-SA 3.0
8 events
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| Mar 6, 2018 at 13:05 | history | edited | j.c. | CC BY-SA 3.0 |
since this is bumped anyways, add latex, add links to references
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| May 16, 2010 at 13:30 | comment | added | David Ben-Zvi | Victor - the Bloch paper is "The dilogarithm and extensions of Lie algebras". Of course K_2 is from its definition all about central extensions, but as far as I know this was the first place it was related to affine Kac-Moody algebras. | |
| May 16, 2010 at 8:05 | comment | added | Torsten Ekedahl | My real point about the Steinberg extension was left out of my comment (because I forgot to make it...), I wanted to speculate that what Bloch did was relating the different kernels. | |
| May 16, 2010 at 6:04 | comment | added | Victor Protsak | @Torsten: 1. Comment about "wandering across Bruhat cells" is right on! 2. It appears that if G is split and simply-connected, the kernel of the universal central extension is Milnor $K_2(k)$ except when G of type C (which includes $SL_2$ in rank 1 case); for type C, it is somewhat larger - see Deligne and Brylinski, numdam.org/numdam-bin/item?id=PMIHES_2001__94__5_0, Introduction. | |
| May 16, 2010 at 5:50 | comment | added | Torsten Ekedahl | ..... Note however that actual loops may wander across Bruhat cells so it doesn't seem to give a complete cocycle in the topological case. | |
| May 16, 2010 at 5:50 | comment | added | Torsten Ekedahl | $K_2$ is defined in terms of the Steinberg extension of $SL_n$ so perhaps the point is that you get a map from the kernel of the universal central extension to $K_2$ for all groups (if I remember correctly even for $SL_n$ and a field these kernels only stabilise from $n=3$ and onwards). In any case going to the cocycle question I think that from the Steinberg point of view it is clear that we get sections over the Bruhat cells and hence (in principle at least) a cocycle desciption for the algebraic loop group. ..... | |
| May 16, 2010 at 5:03 | comment | added | Victor Protsak | Hmmm... I thought that $K_2$ had made its appearance much earlier: according to Milnor in "Algebraic K-theory", work of Steinberg, Moore, and Matsumoto (on universal central extensions of Chevalley groups) was the main motivation for considering the new functor $K_2$. What did Bloch discover then? Deligne's "Symbole modere", on the other hand, only treats the case of commutative G. The difficulty for giving an explicit formula in all cases seems to be that the extension does not split topologically. | |
| May 16, 2010 at 4:53 | history | answered | David Ben-Zvi | CC BY-SA 2.5 |