Timeline for Hopf algebra of Chevalley group from the root system
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 14, 2011 at 13:27 | vote | accept | Baptiste Calmès | ||
| May 18, 2011 at 18:54 | comment | added | Baptiste Calmès | @ Bugs Bunny: I mean the coordinate ring algebra, together with its Hopf algebra structure. @ Carnahan: No, this is not what I want. The language of SGA3 is mostly that of functors of points. What I want is a description of the coordinate rings and their Hopf algebra structure in terms of some elements where only the combinatorics of the root system (or root datum) appears. I'm aware that in principle, one could extract such a description from the functor of points description of SGA3, but I'm actually asking if someone has already done the (non trivial) exercise. | |
| May 18, 2011 at 18:18 | comment | added | mephisto | Make that: any field F of characteristic zero. | |
| May 18, 2011 at 15:42 | comment | added | S. Carnahan♦ | SGA 3 Exp. XXIII and XXV together give constructions of reductive groups from pinned reduced root data over any scheme, including $\mathbb{Z}$. Is that what you want? | |
| May 18, 2011 at 15:37 | answer | added | Marty | timeline score: 4 | |
| May 18, 2011 at 15:22 | comment | added | Bugs Bunny | And all of them can be "unifirmly" constructed from the root system. You'd better say what you want to do with your Hopf algebra and how explicitly you want your construction to be... | |
| May 18, 2011 at 15:21 | answer | added | Victor Petrov | timeline score: 2 | |
| May 18, 2011 at 15:20 | comment | added | Bugs Bunny | You have 3 Hopf algebras monkeying around: the group algebra, the dual group algebra and a Z-form of the functions on the corresponding algebraic groups. There may be even more. Which one do you need? | |
| May 18, 2011 at 14:52 | comment | added | mephisto | There is a uniform construction of the split semisimple Lie algebra attached to root system. The corresponding simply connected semisimple group is that attached to Tannakian category of representations of the Lie algebra. This works over any field. For reductive groups, you have to take a subcategory of representations of the Lie algebra. This all may work over Z, but I haven't seen it written out. | |
| May 18, 2011 at 14:40 | history | asked | Baptiste Calmès | CC BY-SA 3.0 |