Timeline for When does Ext^2 vanish in a category of group representations.
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6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 28, 2013 at 18:13 | comment | added | Dietrich Burde | Yes, II know. I wanted to point this out again, because $L$ need not be semisimple then, but almost. | |
| May 28, 2013 at 18:12 | comment | added | Mariano Suárez-Álvarez | Dietrich, that is precisely the result of Zusmanovich I linked to :-) | |
| May 28, 2013 at 18:11 | comment | added | Dietrich Burde | If $H^2(L,M)=0$ for all finite-dimenional $L$ modules then $L$ is either a $1$-dimensional algebra, or a semisimple algebra, or a direct sum of both (in characteristic zero). | |
| May 28, 2013 at 18:00 | comment | added | Mariano Suárez-Álvarez | Since the usual homological proof of the semisimplicity of semisimple algebras is based on the second Whitehead lemma, one could guess that vanishing of $H^2$ implies semisimplicity. At the level of Lie algebras, this was proved by our fellow MOer Pasha Zusmanovich here, arxiv.org/abs/0704.3864 | |
| May 28, 2013 at 17:52 | comment | added | Mariano Suárez-Álvarez | If $G$ is $k^n$ with its additive group structure and $n\geq2$, then you can easily find $V$ and $W$ such that $\operatorname{Ext}^2(U,V)$ is not zero. | |
| May 28, 2013 at 17:49 | history | asked | Xandi Tuni | CC BY-SA 3.0 |