Timeline for What is the "quaternionic" super Brauer group?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Jan 8, 2016 at 21:35 | comment | added | AHusain | @TheoJohnson-Freyd Thanks. I was thinking of invertible objects of the double $Z ( Rep ( \mathbb{C} \mathbb{Z}_2 ))$ so not thinking of it as symmetric. | |
| Jan 8, 2016 at 16:05 | comment | added | Theo Johnson-Freyd | @AHusain The list $\mathbb Z/2, \mathbb Z/2, \mathbb C^\times$ is correct. The first is the Brauer group $\{\mathbb C,\mathbb Cliff(1)\}$, i.e. the invertible algebras. The second is the Picard group, i.e. the invertible modules. There are two of these: the even and the odd ones. The third is the multiplicative group, i.e. the invertible numbers. | |
| Jan 8, 2016 at 15:59 | comment | added | Theo Johnson-Freyd | Fair enough. I'll check things carefully if I need to. But I would expect some general nonsense about Galois blah blah to provide the claim. | |
| Jan 8, 2016 at 13:04 | history | edited | André Henriques | CC BY-SA 3.0 |
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| Jan 8, 2016 at 11:24 | comment | added | André Henriques | Hi Theo: I do not have any references for that sort of calculation, and I strongly suspect that there are no references for that sort of calculation. I should also point out that my claim that $sBrPic_{\mathbb R}$ is the homotopy fixed points of $sBrPic_{\mathbb C}$ is something that I haven't checked. I just know that the spectral sequence works out, and so it's a plausible statement. | |
| Jan 8, 2016 at 3:54 | comment | added | Theo Johnson-Freyd | And I totally agree that not every symmetric monoidal category should correspond to a version of K-theory. But $\mathrm{SuperVect}_{\mathbb H}$ is the fixed points of a certain $\mathrm{Gal}(\mathbb C/\mathbb R)$-action on $\mathrm{SuperVect}_{\mathbb C}$, which is related to KU-theory, so perhaps the same $\mathrm{Gal}(\mathbb C/\mathbb R)$-action relates $\mathrm{SuperVect}_{\mathbb H}$ to the "fixed points" of KU for that funny action. | |
| Jan 8, 2016 at 3:51 | comment | added | Theo Johnson-Freyd | Hi André! Great answer --- I did not know how to do the spectral sequence calculation. Do you have any references for that sort of calculation? There are various funny things about these stories, for example groups that "really" are $\mathrm{coker}(\mathbb G_m \overset{x \mapsto x^2}\longrightarrow \mathbb G_m)$, whose $\mathbb R$-points are $\mathbb Z/2 \times \mathrm B(\mathbb Z/2)$, but whose $\mathbb C$-points are just $\mathrm B(\mathbb Z/2)$; is it obvious, for example, why $\mathbb Z/2 \times \mathrm B(\mathbb Z/2)$ should be the "$\mathbb Z/2$-fixed point" of $\mathrm B(\mathbb Z/2)$? | |
| Jan 8, 2016 at 3:47 | vote | accept | Theo Johnson-Freyd | ||
| Jan 8, 2016 at 1:17 | comment | added | AHusain | I thought $BrPic(SuperVect_k )$ was $\mathbb{Z}_2 , \mathbb{Z}_2^2 , k^*$. Where is my error? | |
| Jan 8, 2016 at 0:20 | history | edited | André Henriques | CC BY-SA 3.0 |
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| Jan 8, 2016 at 0:12 | history | edited | André Henriques | CC BY-SA 3.0 |
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| Jan 8, 2016 at 0:05 | history | edited | André Henriques | CC BY-SA 3.0 |
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| Jan 7, 2016 at 23:54 | history | edited | André Henriques | CC BY-SA 3.0 |
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| Jan 7, 2016 at 23:49 | history | answered | André Henriques | CC BY-SA 3.0 |