Talk:Killing form

Theorems
The theorems read like definitions. They should be worded in a way which shows one thing implies another.Dewa 19:39, 24 February 2007 (UTC)


 * I add a single sentence to say that the properties are theorems that follow from the definition. 67.198.37.16 (talk) 16:29, 28 October 2020 (UTC)

Properties
Please write in this section instaed of this 'associative' property, which is a non-appropriate name, because this has nothing to do with any kind of associativeness, the equivalent form

B([x,y],z) + B(y,[x,z]) = 0  ,

because this form is the infinitesimal form of the following automorphism equation and has itself a Lie algebraic meaning. It states that left-multiplication in Lie algebras may be seen, besides being an (inner) derivation of the Lie algebra, as a derivation of the Killing form, hence a subalgebra of a pseudo-orthogonal Lie algebra (at least in the semi-simple case). — Preceding unsigned comment added by 130.133.155.66 (talk) 17:42, 15 October 2012 (UTC)


 * Restating it to say that it is a Derivation (differential algebra) would indeed be more appropriate. This is a generic property of all Casimir operators. I have no clue at all about the second half of what you say. I'll try to fix the first half tomorrow, if I can. 67.198.37.16 (talk) 06:08, 28 October 2020 (UTC)
 * Done. 67.198.37.16 (talk) 16:26, 28 October 2020 (UTC)

Matrix elements
Wouldn't it be more appropriate to swap upper and lower indices throughout in this section, especially if $B$ is to function as a metric tensor on $g$? YohanN7 (talk) 10:54, 3 September 2014 (UTC)
 * Yes, it would be. Maybe I'll fix this tomorrow. 67.198.37.16 (talk) 05:52, 28 October 2020 (UTC)
 * Done 67.198.37.16 (talk) 16:34, 28 October 2020 (UTC)

I am relative beginner trying to reproduce the matrix element $$B_{ij}$$. I did it successfully but the sentence "The index k functions as column index and the index n as row index in the matrix $$ad(e_i)ad(e_j)$$. Taking the trace amounts to putting k = n and summing, and so we can write" looks a bit suspicious. It looks like k may be the row index from my calculation? — Preceding unsigned comment added by Penrose sachdev (talk • contribs) 18:09, 7 January 2024 (UTC)