Timeline for Why the Killing form?
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| Oct 12, 2018 at 23:58 | comment | added | TheQuantumMan | @Qfwfq Yes, the analogy is the latter which you commented on. It is only an analogy meant to motivate undergrads. At that level, I can personally not find any intuitive way to think about it. Cheers! | |
| Oct 12, 2018 at 23:48 | comment | added | Qfwfq | Now that I re-read, I realize you were making the analogy with the ordinary dot product on $\mathbb{R}^n$ being "summing over one index" while the Killing becomes "summing over two indices" because "the vectors are matrices". Personally I find this extrinsic viewpoint quite confusing, but maybe other people or the students can make sense out of it... | |
| Oct 12, 2018 at 23:42 | comment | added | Qfwfq | "you can just view the Killing form as the analogue/generalization of the inner product at the tangent space" - the inner product on a tangent space to an abstract manifold? Even in case you think of $G$ as a concrete closed subgroup $\subseteq\mathrm{GL}_n(\mathbb{R})$, the Killing form on $\mathfrak{g}=T_1G$ is not in general that induced from the standard dot product (a.k.a. trace form) on $\mathrm{Mat}_n(\mathbb{R})$, as the latter is always (positive) definite. | |
| Oct 12, 2018 at 21:30 | review | First posts | |||
| Oct 12, 2018 at 22:12 | |||||
| Oct 12, 2018 at 21:25 | history | answered | TheQuantumMan | CC BY-SA 4.0 |