Timeline for Deformation of the Hochschild-Kostant-Rosenberg isomorphism for universal enveloping algebra
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 6, 2019 at 17:26 | comment | added | user108998 | @Adrien ok, that makes sense, I suppose I was being overly optimistic wrt the existence of anything simple. Thanks for the comments. | |
| Mar 6, 2019 at 17:06 | comment | added | Adrien | @EBz indeed, my point was somehow that the corrections you need to add are precisely the corrections you need to go from HKR to Kontsevich, so I doubt there will be a simple formula (basically in the "usual" formality picture you change the map and keep the same differential on both side, while you want to keep the map and change the differential, those are just two ways of doing the same thing). | |
| Mar 6, 2019 at 16:39 | comment | added | Bertram Arnold | The assignment $U\mapsto CE^*(\Omega^*_c(U)\otimes\mathfrak g)$ (tensor with compactly supported forms to get a dgla, then take cohomology) defines a locally constant factorization algebra on manifolds of a given dimension. In particular, $F((a,b))$ is an ($E_1$-)algebra, and Owen Gwilliam proved you get $U\mathfrak g$ this way (people.math.umass.edu/~gwilliam/thesis.pdf Section 4.6). Then $HH_*(U\mathfrak g)\simeq F(S^1)$. To compute the latter, you may replace $\Omega_{(c)}^*(S^1)$ by its subalgebra of harmonic forms, so that $F(S^1)\simeq CE^*(\mathfrak g\ltimes \mathfrak g[1])$. | |
| Mar 6, 2019 at 16:38 | comment | added | user108998 | @Adrien I agree that the existence should be implied by formality, I was hoping for a relatively simple formula for the diff on this. If I'm not mistaken in my calculations the map above doesn't intertwine the hochschild diff and the poisson one, and one needs to add further corrections of lower filt degree. Ofc these must be homologically insignificant as, like u mention, the two sides have the same homology. | |
| Mar 6, 2019 at 16:27 | comment | added | Adrien | I might be missing the point, but isn't it a consequence of Tsygan formality, see eg arxiv.org/abs/math/0010321 ? Ie there is in particular a quasi-iso from HH of $U(\mathfrak g)$ to Poisson homology of $\mathfrak g^*$. | |
| Mar 6, 2019 at 16:08 | history | edited | YCor | CC BY-SA 4.0 |
expanded abbreviation, mathrm Hoch
|
| Mar 6, 2019 at 16:07 | history | edited | user108998 | CC BY-SA 4.0 |
deleted 4 characters in body
|
| Mar 6, 2019 at 16:05 | history | edited | YCor |
edited tags
|
|
| Mar 6, 2019 at 16:01 | history | asked | user108998 | CC BY-SA 4.0 |