Timeline for Why the Killing form?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Feb 25 at 0:48 | comment | added | krm2233 | @VictorProtsak: to play devil's advocate, to some extent that depends on how you state the theorem -- if $(V,\rho)$ is a $\mathfrak g$-representation, and we write $t_V(x,y) = \text{tr}(\rho(x)\rho(y))$, then $\rho(\mathfrak g)$ is solvable if and only if $t_V$ vanishes on $D(\mathfrak g)$. This obviously implies Cartan's Criterion, and it is also potentially more useful in the sense that if a Lie algebra is given as a subalgebra of some $\mathfrak{gl}(V)$, the restriction of the trace form may be simpler to use that the Killing form. | |
| Jul 20, 2010 at 16:17 | comment | added | Victor Protsak | Allen, to state the obvious, Cartan's criterion is extremely useful and fails for other invariant forms. | |
| Jul 20, 2010 at 9:24 | comment | added | Allen Knutson | Are any of the Killing form's extra properties useful? | |
| Jul 20, 2010 at 6:31 | comment | added | Victor Protsak | This seems to be a motivatation for an invariant form on a general representation of $g$, which is a worthy goal, but not for the Killing form per se, which has more to do with the structure of the Lie algebra itself and also has properties not shared by all invariant forms. | |
| Jul 20, 2010 at 4:13 | vote | accept | Ryan Reich | ||
| Jul 20, 2010 at 2:03 | comment | added | Ryan Reich | This is a nice idea! I did actually spend a lecture on finite groups that I can draw on for motivation, and I made a point of talking up the averaging procedure in the course of proving Maschke's theorem. In a better-structured course I would have done the inner product at that time, and then what you say would be perfect. Perhaps I can still rig it... | |
| Jul 20, 2010 at 1:55 | history | answered | David Jordan | CC BY-SA 2.5 |