Timeline for Adjoints of exact functors between semisimple abelian categories
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 1, 2021 at 1:02 | comment | added | Theo Johnson-Freyd | Note that along the way we just proved the Peter–Weyl theorem. | |
| Nov 1, 2021 at 1:02 | comment | added | Theo Johnson-Freyd | Take $W$ simple. Using semisimplicity, you can see that $\hom_{\mathfrak{sl}(2)}(V,W)$ will be of the same dimension as the number of copies of $W$ in a direct sum decomposition of $V$. In other words, if there is an object $V$ with this defining property, then it will need to be $V \overset?= \bigoplus_I \mathrm{dim}(I) I$, where the sum ranges over isomorphism classes of simple objects $I$. Since we know that there are infinitely many non-isomorphic simple $\mathfrak{sl}(2)$, this sum diverges, and there is no such $V$. | |
| Nov 1, 2021 at 0:58 | comment | added | Theo Johnson-Freyd | Ok, let's try to show that $\mathrm{Forget}: \mathrm{Mod}(\mathfrak{sl}(2)) \to \mathrm{Vec}$ fails to have a left adjoint (where both $\mathrm{Mod}$ and $\mathrm{Vec}$ mean the finite-dimensional versions). Well, since the codomain is generated under direct sums by $\mathbb{C}^1$, and since every additive functor preserves direct sums, the question turns on whether there is or isn't an object $V := \mathrm{Forget}^L(\mathbb{C})$. The defining property of this object $V$, from unpacking the adjunction, is $\hom_{\mathfrak{sl}(2)}(V, W) = \mathrm{Forget}(W)$ for every finite-dimensional $W$. | |
| Oct 31, 2021 at 15:22 | history | edited | rvk | CC BY-SA 4.0 |
added 1047 characters in body
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| Oct 31, 2021 at 15:21 | comment | added | Tim Montegue | Thanks a lot. This answer was very helpful. | |
| Oct 31, 2021 at 15:21 | vote | accept | Tim Montegue | ||
| Oct 31, 2021 at 14:53 | history | answered | rvk | CC BY-SA 4.0 |