Timeline for If C is a fusion category over a field of nonzero characteristic and dim C = 0, is Z(C) ever fusion?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 1, 2014 at 10:30 | vote | accept | Noah Snyder | ||
| Jan 4, 2014 at 22:07 | answer | added | Noah Snyder | timeline score: 7 | |
| S Oct 29, 2013 at 18:12 | history | suggested | Manuel Bärenz | CC BY-SA 3.0 |
Improved TeX formatting
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| Oct 29, 2013 at 18:03 | review | Suggested edits | |||
| S Oct 29, 2013 at 18:12 | |||||
| Jun 8, 2013 at 4:23 | comment | added | Noah Snyder | I think we've worked out that Z(C) fusion implies $\mathrm{dim} C \neq 0$. I will post an answer soon when the proof is posted. | |
| Aug 11, 2011 at 19:01 | comment | added | Evan Jenkins | That said, your desired result is probably less hard than proving that all fusion categories are spherical. Drinfeld, Gelaki, Nikshych, and Ostrik have managed to prove quite a lot about not-necessarily-spherical categories (including that the center of a fusion category is nondegenerate, the nonspherical analogue of modular), albeit only over an algebraically closed field of characteristic zero. One might hope that combining the BV and DGNO viewpoints would give such a result for nonspherical categories with noninvertible dimensions, but I don't want to speculate too much about it. | |
| Aug 11, 2011 at 18:53 | comment | added | Evan Jenkins | It appears so. In the paper "Categorical centers and Reshetikhin-Turaev invariants," Bruguieres and Virelizier show that the center of a spherical category is (not necessarily semisimple) modular even when the dimension is noninvertible, although it seems that some essential pieces of the proof are relegated to a paper that is still forthcoming. | |
| Aug 11, 2011 at 6:23 | comment | added | Noah Snyder | That means any counterexample would have to be nonspherical, right? Hence there couldn't be any known counterexamples. (Or maybe I'm missing something about what modularity means in nonzero characteristic.) Probably that can be bumped up to a proof by looking at the sphericalization (at least outside of characteristic 2). | |
| Aug 11, 2011 at 5:56 | comment | added | Evan Jenkins | This is far from an answer, but on page 3 of "From Subfactors to Categories and Topology II," Mueger suggests that "there is little hope of proving semisimplicity" of the center when the dimension is 0. Of course, this does nothing to rule out the possibly that the center might just happen to be semisimple in some anomalous cases. He also points out that in any case the center could never be modular, as modular categories necessarily have nonzero dimension. | |
| Aug 11, 2011 at 4:19 | history | asked | Noah Snyder | CC BY-SA 3.0 |