Timeline for What algebraic group does Tannaka-Krein reconstruct when fed the category of modules of a non-algebraic Lie algebra?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Jul 9, 2010 at 4:56 | vote | accept | Theo Johnson-Freyd | ||
| Apr 15, 2010 at 15:53 | comment | added | Theo Johnson-Freyd | @JSMilne: Fair enough. I've only actually seen the reconstruction computed all the way through for the semisimples. You should go through and ignore the word "non-algebraic" throughout the question. Or, better, remember that this is the "mathematician's non": the set "non-algebraic Lie algebras" includes the set "algebraic Lie algebras", just like "noncommutative rings" includes "commutative rings". | |
| Apr 15, 2010 at 15:51 | comment | added | Theo Johnson-Freyd | @QY: Absolutely. But notice that for Tannaka-Krein to reconstruct an actual group puts restrictions on the category of modules: it must be symmetric, and the forgetful functor must preserve the symmetry. From the Hopf algebraic perspective, this is the requirement that H = U(g) is cocommutative, and then TK will reconstruct some dual to H, which will be a commutative algebra and hence a scheme. But anyway, if H is cocommutative, it's pretty close to a universal enveloping algebra already. | |
| Apr 15, 2010 at 13:27 | answer | added | Torsten Ekedahl | timeline score: 22 | |
| Apr 15, 2010 at 7:47 | answer | added | JS Milne | timeline score: 32 | |
| Apr 15, 2010 at 6:15 | comment | added | JS Milne | There seems to be an assumption implicit in the question that when you start with an algebraic Lie algebra g, the affine group scheme G you get has Lie algebra g. As @unknown noted, this is far from true (except for semisimple Lie algebras in characteristic zero). So before trying to understand the affine group schemes you get from nonalgebraic Lie algebras, perhaps you should try to understand those you get from algebraic Lie algebras (e.g., the one-dimensional Lie algebra). | |
| Apr 15, 2010 at 5:56 | comment | added | Qiaochu Yuan | Since representations of g can be identified with modules over U(g) perhaps the question should be generalized to Hopf algebras? | |
| Apr 15, 2010 at 5:05 | comment | added | naf | It might be something very big since that is so even for some Lie algebras which are Lie algebras of algebraic groups. For example, let $\mathfrak{g}$ be a 1-dimensional Lie algebra. Even though it is the Lie algebra of an algebraic group it seems to me that $\mathcal{G}$ in this case is infinite dimensional. | |
| Apr 15, 2010 at 4:09 | comment | added | Mariano Suárez-Álvarez | Very nice question. | |
| Apr 15, 2010 at 4:06 | history | asked | Theo Johnson-Freyd | CC BY-SA 2.5 |