Timeline for Can we see the geometric realization of $U_q(sl_2)$'s relations as Schubert Conditions?
Current License: CC BY-SA 2.5
10 events
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| Oct 29, 2010 at 20:55 | vote | accept | B. Bischof | ||
| Oct 29, 2010 at 17:11 | comment | added | Ben Webster♦ | Nakajima is exactly talking about the BLM construction; there's no real significance to the fact that Nakajima uses finite dimensional things in his paper and BLM use infinite dimensional. You really just take the limit as N goes to infinity. If you don't take this limit, you only get the quotient of sl(2) that only acts faithfully on representations of highest weight $\leq N$, so you have to send $N$ to infinity to get all of the algebra. | |
| Oct 29, 2010 at 17:05 | answer | added | Ben Webster♦ | timeline score: 1 | |
| Oct 29, 2010 at 16:39 | comment | added | Jim Humphreys | @Bischof I'm definitely not an expert on these things, but keep in mind that quantum groups (except at a root of unity) tend to have an infinite dimensional flavor in terms of representations and such. So anything like Schubert conditions here would likely have that flavor. | |
| Oct 29, 2010 at 16:27 | history | edited | B. Bischof | CC BY-SA 2.5 |
added a little bit.
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| Oct 29, 2010 at 16:26 | comment | added | B. Bischof | @Jim Well, Nakajima also mentions the extension of this to $\mathfrak{sl}_n$ which I am actually more interested in. Further from that line that you quote, it sounds even more like a Schubert condition. The inf dim part worries me a bit however. Thanks for your comments. | |
| Oct 29, 2010 at 13:51 | comment | added | Jim Humphreys |
This Duke J. paper may require the help of a library to access online, but one line from the introduction sums it up: "In this paper we construct the entire algebra (not only the + part) assuming that we are in type A, using the geometry of relative positions of pairs of flags in infinite dimensional space." This may not be close enough to Nakajima's formulation involving $\mathfrak{sl}_2$ to be helpful.
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| Oct 29, 2010 at 13:31 | comment | added | Jim Humphreys | @Bischof This is a 1990 Duke J. paper: MR1074310 (91m:17012). Be˘ılinson, A. A. [Beilinson, Aleksandr A.] (1-MIT); Lusztig, G. (1-MIT); MacPherson, R. [MacPherson, RobertD.] (1-MIT). A geometric setting for the quantum deformation of GLn. Duke Math. J. 61 (1990), no. 2, 655–677. | |
| Oct 29, 2010 at 7:27 | comment | added | B. Bischof | This might be spoken about in "a geometric setting for quantum groups" by Beilinson, Lusztig, and MacPhearson, but I can't seem to get a copy to check. | |
| Oct 29, 2010 at 7:05 | history | asked | B. Bischof | CC BY-SA 2.5 |