Timeline for Poisson algebras as deformations vs. Poisson algebras in algebraic topology
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| Sep 13, 2011 at 11:47 | comment | added | DamienC | @Theo: you are right! I actually think that you get a Drinfeld associator stricto sensu. Tamrakin$^{-1}$ can be guessed from the paper of Severa and Willwacher (arxiv.org/abs/0905.1789), where they compare the comstruction of Tamarkin and the one of Kontsevich. | |
| Sep 12, 2011 at 17:19 | comment | added | Theo Johnson-Freyd | Or I think you can run Tamarkin's associator->formality proof in reverse, and at least get a some slightly homotopy-ized version of Drinfeld associator. | |
| Sep 12, 2011 at 17:18 | comment | added | Theo Johnson-Freyd | @DamienC: Well, erm, I don't remember the literature well enough. Tamarkin certainly shows that having an associator gives a proof of formality when d=2. Conversely, he shows that any proof of formality gives a Lie bialgebra quantization theorem. But that's really not very far from an associator, maybe without some side conditions; in particular, a sufficiently "universal" quantization certainly includes an associator, by an old result recently written up by Bar-Natan and Dancso. | |
| Sep 12, 2011 at 9:46 | comment | added | DamienC | The paper of Severa is arxiv.org/abs/0902.3576. He wrote another relevant paper with Willwacher, about formality when $d=2$: arxiv.org/abs/0905.1789 By the way, did Tamarkin really prove that formality for $d=2$ is the same as existence of Drinfeld associators ? | |
| Sep 12, 2011 at 4:25 | comment | added | Theo Johnson-Freyd | Actually, noncontractibility is also nuanced. When $d=2$, there is a "properad" that presents the notion of "finite-dimensional Lie bialgebra", and you can pose the quantization problem for these, and there is a scheme of quantizations, and a map from the scheme of Drinfeld associators to this scheme. Etingof and Enriquez show that this map has image just one point. But they do not, so far as I know, show that the scheme of "finite dimensional quantizations" is contractible; just that all "possibly-infinite dimensional quantizations" act the same on finite-dimensional Lie bialgebras. | |
| Sep 12, 2011 at 4:23 | history | answered | Theo Johnson-Freyd | CC BY-SA 3.0 |