Timeline for Can one show the equivalence of the abstract and classical Jordan decompositions for simple Lie algebras without complete reducibility?
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| Sep 15, 2013 at 18:18 | comment | added | Will Sawin | Well for a one-dimensional Lie algebra, the universal enveloping algebra is $\mathbb C[x]$, the completion is $\prod_{\lambda \in \mathbb C} \mathbb C[[x-\lambda]]$. $x_s$ and $x_n$ are well-defined in this algebra, but primitivity fails. I think the primitivity criterion works for elements of the universal enveloping algebra but can fail for its completion? Then the semisimplicity has to be used in showing that the Jordan decomposition lies in the universal enveloping algebra, not just its completion. | |
| Sep 15, 2013 at 15:10 | comment | added | Terry Tao | This looks like a very appealing argument, but I wonder how the hypothesis that ${\mathfrak g}$ is simple (or semisimple) comes into this, since of course the theorem is false in general (e.g. for one-dimensional Lie algebras). | |
| Sep 15, 2013 at 14:20 | history | answered | Ben Webster♦ | CC BY-SA 3.0 |