Timeline for Is $G/T$ a projective variety?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Dec 7, 2014 at 15:49 | history | suggested | Ali Taghavi | CC BY-SA 3.0 |
I add a few words
|
| Dec 7, 2014 at 15:37 | review | Suggested edits | |||
| S Dec 7, 2014 at 15:49 | |||||
| Oct 30, 2014 at 15:43 | vote | accept | CommunityBot | ||
| Feb 2, 2014 at 14:30 | comment | added | Jim Humphreys | @Daniel: My offhand reference to BGG here is unhelpful. Instead, the theorem on affine quotients in arbitrary characteristic (and its history) is found in R.W. Richardson's paper Affine coset spaces of reductive algebraic groups, Bull. London Math. Soc. 9 (1977), no. 1, 38–41. For complex semisimple groups (which are Lie groups), this goes back to Matsushima and others. In any case, notation $G/T$ needs context. | |
| Feb 1, 2014 at 23:42 | comment | added | Daniel Pomerleano | @JimHumphreys Do you have a precise reference for this paper of Bernstein, Gelfand and Gelfand? | |
| Feb 1, 2014 at 23:14 | comment | added | Jim Humphreys | @Hassan: It's easy to answer your question if you make clear at the outset whether your "Lie group" has a structure of algebraic variety (say over the complex field). For a semisimple algebraic group $G$ over $\mathbb{C}$, the algebraic variety $G/T$ for a maximal algebraic torus $T$ is definitely not projective but instead affine. (Bernstein-Gelfand-Gelfand wrote a long paper about this variety.) On the other hand, the real manifold $G/T$ for a compact Lie group $G$ is certainly compact, and is related indirectly to a complex projective flag variety $G/B$ as in the Borel-Weil theorem. | |
| Feb 1, 2014 at 20:26 | comment | added | Denis Nardin | I am not sure what do you mean by "is a projective variety". If your group is algebraic there is a natural algebraic variety structure on its quotients, otherwise I do not know. I think you are correct that any complex semisimple Lie group has an algebraic group structure but I'm far from an expert of the subject and I don't want to say something that might be false or misleading. | |
| Feb 1, 2014 at 19:38 | answer | added | Peter Crooks | timeline score: 7 | |
| Feb 1, 2014 at 19:21 | vote | accept | CommunityBot | ||
| Oct 30, 2014 at 15:43 | |||||
| Feb 1, 2014 at 19:20 | comment | added | user21574 | We can not define Borel or parabolic subgroups on non-algebraic groups?. also complex semisimple lie groups can not be viewed as complex linear algebraic groups? | |
| Feb 1, 2014 at 19:11 | comment | added | Denis Nardin | In the theory of algebraic groups subgroups $H$ such that $G/H$ is a complete variety are called parabolic subgroups. Borel subgroup are characterized as being minimal parabolic subgroup, so in general maximal tori won't be parabolic. See e.g. Springer, Linear Algebraic Groups. | |
| Feb 1, 2014 at 19:10 | history | edited | Ben McKay | CC BY-SA 3.0 |
spelling
|
| Feb 1, 2014 at 19:09 | answer | added | Ben McKay | timeline score: 10 | |
| Feb 1, 2014 at 19:05 | history | asked | user21574 | CC BY-SA 3.0 |