Timeline for Resolution of a free lie algebra as a module over its universal enveloping algebra.
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 1, 2010 at 19:54 | comment | added | Don Stanley | I see a problem since the internal differential in the bar construction could land in terms in $T(V)\otimes T(V)\otimes T(V)$ that are not in $T(V)\otimes V\otimes T(V)$. For example if $V$ is free on basis $a,b$ and $db=a^2$. | |
| Mar 1, 2010 at 19:50 | vote | accept | Don Stanley | ||
| Mar 1, 2010 at 19:50 | |||||
| Mar 1, 2010 at 19:44 | vote | accept | Don Stanley | ||
| Mar 1, 2010 at 19:50 | |||||
| Mar 1, 2010 at 14:54 | comment | added | Mariano Suárez-Álvarez | Indeed. The complex is just a subcomplex of the bar construction. | |
| Mar 1, 2010 at 14:45 | comment | added | Ben Webster♦ | Presumably the just the usual differential on the tensor product of compelxes. | |
| Mar 1, 2010 at 14:13 | comment | added | Don Stanley | What is the differential on the $T(V)\otimes V\otimes T(V)$ (or $T(V)\otimes V\otimes L(V)$) term? | |
| Mar 1, 2010 at 14:05 | comment | added | Mariano Suárez-Álvarez | You can take the total complex of the resolution I constructed above: that is a free $T(V)$-module with a differential. The map induced by my map $T(L)\otimes L(V)\to L(V)$ is then a quasi-iso. | |
| Mar 1, 2010 at 13:58 | comment | added | Don Stanley | Slightly less good would be to have a $T(V)$ dg-module quasi-isomorphic to $L(V)$ that was free as a $T(V)$ module (like the bar construction). | |
| Mar 1, 2010 at 13:53 | comment | added | Don Stanley | For the differential I mean that $T(V)$ is a differential graded algebra and $L(V)$ is a differential graded module over it. Then it would be great to have $L(V)$ as the cone on a map in the category of $T(V)$ modules (as in your resolution), but I guess that's not possible. | |
| Mar 1, 2010 at 13:39 | comment | added | Don Stanley | This is great. Thanks a lot. I need to think about it for a bit. | |
| Mar 1, 2010 at 13:26 | history | edited | Mariano Suárez-Álvarez | CC BY-SA 2.5 |
added 195 characters in body
|
| Mar 1, 2010 at 13:21 | history | answered | Mariano Suárez-Álvarez | CC BY-SA 2.5 |