Timeline for What is the universal enveloping algebra?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 2, 2012 at 16:12 | history | edited | David White | CC BY-SA 3.0 |
Fixed some typos
|
| Jun 15, 2010 at 10:44 | vote | accept | Bugs Bunny | ||
| Jun 15, 2010 at 10:44 | comment | added | Bugs Bunny | To celebrate one month's anniversary I am accepting this as an answer, although I still believe it is not general enough... | |
| May 21, 2010 at 9:02 | comment | added | Bugs Bunny | I am still thinking but still remain in the dark, at least in my head. I can write quite universal categories of the sort I am asking by generators and relations. They will have functors to abelian categories but how do I pull info back? Deligne-Morgan is helpful as a formula but the proof is still "abelian". Essentially, I think that one needs "diagrammatic" proof of associativity and universality, akin to Bar-Natan's proof of Duflo Conjecture... Ain't I a stinker? | |
| May 18, 2010 at 19:03 | history | edited | Torsten Ekedahl | CC BY-SA 2.5 |
Complete revision.
|
| May 18, 2010 at 15:47 | comment | added | Theo Johnson-Freyd | This construction does give an associative algebra object in any category over $\mathbb Q$, by the Deligne-Morgan reference above. The trick is to read the formulas like the one Torsten gave above for $u\odot v \odot w$ not as formulas for actual elements (in which case it would be incorrect e.g. in super vector spaces), but as ("multilinear") maps in your category. This means, in particular, that you are not allowed to duplicate or delete variables in your formulas, a restriction that Baez calls "quantum". Once you restrict yourself, whatever symbolic argument you want to run you can. | |
| May 18, 2010 at 9:44 | comment | added | Bugs Bunny | Thank you very much. This is excellent! However, I do not see how one can reduce proving associativity and universality to vector spaces. Suppose we have defined a multiplication $\mu:S({\mathfrak g})\otimes S({\mathfrak g})\rightarrow S({\mathfrak g})$. Can we actually apply the functor $hom(I, )$ to this map? Is this functor tensor? | |
| May 17, 2010 at 19:20 | history | edited | Torsten Ekedahl | CC BY-SA 2.5 |
Added comment on universal property.
|
| May 17, 2010 at 19:06 | history | answered | Torsten Ekedahl | CC BY-SA 2.5 |