Timeline for Tannakian reconstruction and the distribution algebra
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 1, 2023 at 8:58 | comment | added | Adrien | I'm sure you know Cartier's Primer on Hopf algebras, this is there in a different language I think. Then if $\phi \in End(F)$, define $$\bar \Delta(\phi)_{V,W}=\phi_{V\otimes W}-\phi_V-\phi_W$$ it might be not too hard using this to show that $\phi$ vanishes on $m^n$ iff $\bar\Delta^n(\phi)=0$ which I think it's exactly the result you conjectured. I feel like this should be fairly standard but can't think of a reference right now.. | |
| Jul 1, 2023 at 8:55 | comment | added | Adrien | To me this is the statement that $O(G)$ is coendomorphisms of the fiber functor. Explicitly, if $k$ is a field, if $C=Rep_G$, $O(G)=\int^C x^* \otimes x$ (if $k$ is a ring probably you can take the coend over comodule which are say free of finite rank so that the dual makes sense). On the other hand $End(F)=\int_C End(x)=\int_C x^*\otimes x$, and then use that $Hom(-,k)$ turns colimits into limits. You can also seetthis more abstractly using left/right adjoints to $F$. | |
| Jul 1, 2023 at 2:22 | comment | added | Antoine Labelle | That seems plausible, at least there is a map from $\text{Hom}(\mathcal{O}(G)), k)$ to $\text{End}(F)$ and it is an iso in the example $G=\mathbb{G}_m$. Do you have an idea of how to prove that or a reference? | |
| Jun 30, 2023 at 8:48 | comment | added | Adrien | Right, though I might be wrong but isn't $End(F)$ literally just $Hom(O(G),k)$ ? | |
| Jun 29, 2023 at 13:40 | comment | added | Antoine Labelle | Yes, you can do that of course but I want a more direct way to reconstruct the distribution algebra, specifically via its map to $End(F)$. | |
| Jun 29, 2023 at 8:15 | comment | added | Adrien | I presume this is not what you want, but why not reconstruct $O(G)$ from $Rep_G$ instead and then define $Dist(G)$ as you did ? | |
| Jun 28, 2023 at 19:57 | comment | added | LSpice |
It seemed unlikely that you really meant $\text{Rep}\_G$ \text{Rep}\_G when you used $\text{Rep}_G$ \text{Rep}_G earlier, so I edited accordingly. I also changed the various \texts to \operatornames.
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| Jun 28, 2023 at 19:55 | history | edited | LSpice | CC BY-SA 4.0 |
TeX fixes; name of MSE question
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| Jun 28, 2023 at 19:28 | history | asked | Antoine Labelle | CC BY-SA 4.0 |