Timeline for Derived category of representations
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Dec 29, 2013 at 20:33 | comment | added | Jim Humphreys | @Aleksa: Jantzen's book covers much of the modular representation theory. Derived categories come up mostly when you translate into algebraic geometry, as people like Bezrukavnikov and Mirkovic have done in their papers (see arXiv). It does get complicated. | |
| Dec 29, 2013 at 15:25 | comment | added | Sasha | The book of Jantzen is a good starting point. | |
| Dec 29, 2013 at 15:09 | comment | added | Aleksa | That's nice. Can you please give me some titles of the literature? | |
| Dec 29, 2013 at 15:02 | comment | added | Jim Humphreys | @Aleksa: In characteristic 0 it's usually simpler just to study one block of representations at a time. On the other hand, for reductive groups in prime characteristic (not linearly reductive), the module categories get much more interesting and there are injectives, tilting objects, etc. In either situation there is a lot of literature out there. | |
| Dec 29, 2013 at 14:55 | comment | added | Sasha | If it is a complex with finite number of finite-dimensional cohomology, still there are only finitely many simple summands in those and everything missing is in the orthogonal. | |
| Dec 29, 2013 at 14:47 | comment | added | Aleksa | Do you assume the tilting object is a finite dimensional representation...I mean the tilting object could be a komplex in $D^b(Repr(G))$... So what is when $G$ is not assumed to be linearly reductive? | |
| Dec 29, 2013 at 14:05 | comment | added | Sasha | If the number of simple objects is infinite, then there is no finite dimensional tilting generator (for any finite dimensional representation there is a simple object which is not its summand and so it is in the orthogonal). | |
| Dec 29, 2013 at 12:58 | comment | added | Aleksa | Ok. But the number of these simple representations may be infinite right? Are there some nontrivial tilting objects known for certain groups? | |
| Dec 29, 2013 at 12:40 | comment | added | Sasha | The category is semisimple --- just take the sum of all simple representations. | |
| Dec 29, 2013 at 12:38 | history | asked | Aleksa | CC BY-SA 3.0 |