Timeline for Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 4, 2018 at 10:57 | vote | accept | Bas Winkelman | ||
| Nov 4, 2018 at 9:37 | comment | added | Adrien | So @NoahSnyder I'd say this is 1), classifying formal deformations is definitely easier, inequivalent $q$-deformations could potentially becomes equivalent as formal ones. | |
| Nov 4, 2018 at 9:31 | comment | added | Adrien | @BasWinkelman Compare with the statement that every formal deformation of $U(\mathfrak g)$ is isomorphic to $U(\mathfrak g)[[\hbar]]$ as an algebra (for $\mathfrak g$ simple). There isn't an analogous statement for the $q$ version. Now any deformation over $\mathbb C[q,q^{-1}]$ embeds as an algebra into $U[[\hbar]][\hbar^{-1}]$ but you definitely cannot conclude that any two $q$-deformations are equivalent in any sense. So the $q$-version is definitely more subtle. In particular you really need to work at the categorical level, and to restrict to (locally) finite-dimensional ones. | |
| Nov 4, 2018 at 0:45 | comment | added | Noah Snyder | That's a good question. I'd think that the tangent space to the space of q-deformations has an injection to the space of h-deformations, which would say that the former is bounded if you know the latter... The results that I quoted are actually stronger than just saying you know the q-deformations. So there are two options: 1) I'm wrong and just to classify deformations is easier, or 2) there's something subtle going on like maybe the h-adic results assume something more (e.g. that the fiber functor also deforms). | |
| Nov 3, 2018 at 20:52 | comment | added | Bas Winkelman | I care more about the $q$-deformations, so thanks you for your answer. As Adrien has pointed out in his answer, the question seems to be settled for $h$-adic case, so why is it not possible to "transfer" the proof to the $q$-case? | |
| Nov 3, 2018 at 18:17 | history | answered | Noah Snyder | CC BY-SA 4.0 |