Timeline for What is Koszul duality?
Current License: CC BY-SA 2.5
19 events
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| Jun 22, 2022 at 7:16 | history | edited | CommunityBot |
replaced http://front.math.ucdavis.edu/ with https://arxiv.org/abs/
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| Aug 25, 2021 at 6:58 | comment | added | მამუკა ჯიბლაძე | @MartinSleziak Thank you very much, Martin! Extremely helpful, as always | |
| Aug 25, 2021 at 5:36 | comment | added | Martin Sleziak | @მამუკაჯიბლაძე At least some version can be found in the Wayback Machine. (I will add also arXiv link from the answer - since the front end is down for some time.) | |
| May 5, 2017 at 9:59 | comment | added | მამუკა ჯიბლაძე | @HeinrichHartmann could you revive the link? Or has it been removed on purpose? | |
| Jun 15, 2012 at 20:24 | comment | added | Heinrich Hartmann | In an effort to understand this answer I worked out some detais, gathered references, gave a seminar talk and puta all of this on my webpage: heinrich-hartmann.net/wiki/index.php/Koszul_Duality. | |
| Sep 22, 2010 at 13:10 | comment | added | skupers | "Another (closely related) example is that the cohomology of a reductive group (over C) is Koszul dual to the cohomology of its classifying space (here you see a hint of this delooping mentioned in Scott's answer)." -- Does anyone know of a reference for this? It sounds really interesting and a good motivation to learn more about Koszul duality. | |
| Nov 18, 2009 at 19:16 | history | edited | Ben Webster♦ | CC BY-SA 2.5 |
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| Oct 19, 2009 at 17:38 | comment | added | Ilya Nikokoshev | "Formality in the way I and Ben use the word means there is an A-infinity structure on the homology of a dg-structure with all higher (>2) operations trivial." -- yes, the same one, but I'm not an expert to explain more. | |
| Oct 19, 2009 at 17:37 | vote | accept | Ilya Nikokoshev | ||
| Oct 14, 2009 at 9:39 | comment | added | James Griffin | That's certainly the notion of formality I'm familiar with. Kontsevich formality concerns just one particular class of examples: the Hochschild cohomology of a "smooth algebra" has a dg-Lie algebra structure. Kontsevich formality says that it's formal. | |
| Oct 13, 2009 at 23:24 | comment | added | Mikael Vejdemo-Johansson | Formality in these terms is, I think, related to Kontsevich formality. Formality in the way I and Ben use the word means there is an A-infinity structure on the homology of a dg-structure with all higher (>2) operations trivial. I am slightly too unfamiliar with Kontsevich formality or the Kontsevich framework to talk reliably about what the relation really is. | |
| Oct 13, 2009 at 20:49 | comment | added | Ben Webster♦ | I don't know what Kontsevich formality is, so you don't need to know anything about that to understand what's going on. I just mean that the fact that A-infinity products all preserve internal grading, and change homological grading means that they all have to vanish if these coincide. Another way to think about this it to take a dg-model for the Ext-algebra. If we separate this into its various internal degrees, we get a bunch of complexes with homology in exactly one degree, so we can pick any quasi-isomorphism we like on each internal degree, and see that the alegbra is formal. | |
| Oct 13, 2009 at 19:57 | comment | added | Ilya Nikokoshev | " trivial (formal) A-infinity structure for relatively simple degree reasons" --- is that somehow about Kontsevich formality? | |
| Oct 13, 2009 at 19:57 | comment | added | Ilya Nikokoshev | @Mikael Vejdemo-Johansson: Yes! Please do write up an answer of your own. Absolutely would be great talk about operads and I think this is how Koszulity is related to derived alg-geom? | |
| Oct 13, 2009 at 19:55 | vote | accept | Ilya Nikokoshev | ||
| Oct 13, 2009 at 19:58 | |||||
| Oct 13, 2009 at 4:58 | comment | added | Ben Webster♦ | That's fair. I'm just trying to bring in the perspective that tends to get used in geometric representation theory, since that's what ilya had originally been asking about. Of course, it would be good to hear your spiel too. | |
| Oct 13, 2009 at 3:56 | comment | added | Mikael Vejdemo-Johansson | Well... There is more to Ext(L,L) than just its A-infinity structure, and there is more to Koszul duality being nice than that it forces a trivial (formal) A-infinity structure for relatively simple degree reasons. As I view it, the A-infinity side of Koszulity is mostly Yet Another Good Thing it brings. I should write up an answer of my own here. One of these days. :-) Probably talking at length about operads, and about how to find Nice resolutions of things. | |
| Oct 12, 2009 at 23:58 | history | edited | Ben Webster♦ | CC BY-SA 2.5 |
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| Oct 12, 2009 at 23:02 | history | answered | Ben Webster♦ | CC BY-SA 2.5 |