Timeline for Why do Lie algebras pop up, from a categorical point of view?
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6 events
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| Jul 31, 2014 at 2:50 | comment | added | Todd Trimble | @PeterLeFanuLumsdaine I found it helpful to consider that the universal enveloping algebra $U: \text{LieAlg} \to \text{HopfAlg}$ has a right adjoint given by taking primitive elements $P: \text{HopfAlg} \to \text{LieAlg}$, and that in characteristic $0$ the unit $1 \to P U$ of the adjunction is an isomorphism. This implies that $U$ is fully faithful. Milnor-Moore identifies those Hopf algebras $H$ where the counit is an isomorphism. | |
| Jul 25, 2014 at 18:27 | comment | added | Qiaochu Yuan | @Peter: as Sinan says, the extra condition is called conilpotence. See Theorems 3.6.1 and 3.8.1 in Cartier's A primer of Hopf algebras (people.math.osu.edu/kerler.2/VIGRE/InvResPres-Sp07/…). The underlying field needs to have characteristic $0$. | |
| Jul 25, 2014 at 16:38 | comment | added | Sinan Yalin | In the setting of (differential) graded modules, this comes from the Milnor-Moore theorem, but you have to add a conilpotence condition on your Hopf algebras to get the equivalence of categories. For instance, you can take a look at the chapter Hopf algebras, Theorem 7.2.19, which states a version slightly more general than the original result. | |
| Jul 25, 2014 at 12:29 | comment | added | Peter LeFanu Lumsdaine | “The category of Lie algebras is equivalent to a certain category of cocommutative Hopf algebras” — do you know a good reference that works this out or states it more precisely? Or at least some specific term which readers can google for more details? | |
| Jul 25, 2014 at 10:09 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
added 667 characters in body
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| Jul 25, 2014 at 9:56 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |