Timeline for Can the ribbon category of f.d. reps of $\mathcal{U}_q(\mathfrak{sl}(2))$ be modified so the twist is trivial on the vector representation?
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9 events
| when toggle format | what | by | license | comment | |
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| Feb 6, 2016 at 17:06 | comment | added | Theo Johnson-Freyd | @NoahSnyder Ah, yes, that's much clearer. | |
| Feb 6, 2016 at 17:05 | comment | added | Theo Johnson-Freyd | @AndyManion Just the vector. Already for the symmetric and exterior squares, the ribbon is something like multiplication by $q$. | |
| Feb 5, 2016 at 23:43 | vote | accept | Andy Manion | ||
| Feb 4, 2016 at 23:49 | comment | added | Noah Snyder | The point here is roughly that central character gives a grading on representations, and so elements of the center of the corresponding Lie group give you the ways of changing ribbon structure. For SL the center is finite and so you only get a few ribbon structures, but for GL the center is the scalars and so you can change ribbon structure at the vector rep however you want. The ribbon structure won't be trivial everywhere, in particular the quantum determinant will no longer have trivial ribbon element (which is why this ribbon structure doesn't descend to SL). | |
| Feb 4, 2016 at 21:04 | comment | added | Andy Manion | For the category you describe in the first paragraph: is the ribbon element trivial everywhere, or just on the defining object? | |
| Feb 4, 2016 at 16:46 | comment | added | Theo Johnson-Freyd | Come to think of it, I said "quotient out by the ideal of negligible morphisms", which I do think always works, but I'm going off of memory, and one regularly simplifies claims in memory that are more complicated in real life. If so, then one version of the "local relations" question is whether that ideal is finitely generated. | |
| Feb 4, 2016 at 16:42 | comment | added | Theo Johnson-Freyd | ... Schur–Weyl duality recently, including by regulars here on MO, which perhaps has answered the question, but I admit I have not kept up to date on the results. | |
| Feb 4, 2016 at 16:41 | comment | added | Theo Johnson-Freyd | I should warn: I'm not sure how much of my representation theory claims are on firm footing. Indeed, my impression is that there are (or at least recently were) open questions about whether you actually produce $\mathcal U_q(\mathfrak{gl}(n))$ from following a procedure like I said, or just some category that maps to it. The question is: "is $\mathrm{Rep}(\mathcal U_q\mathfrak g)$ described by local relations"? The Kuperberg spiders give the answer "yes" for the $\mathfrak g$ of rank-2, and for a long time it was open in higher rank. I know that there's been major recent work on quantum... | |
| Feb 4, 2016 at 16:37 | history | answered | Theo Johnson-Freyd | CC BY-SA 3.0 |