Let $\mathfrak{g}$ be a semi-simple Lie algebra and let $\phi:\mathfrak{g}\rightarrow\mathfrak{gl}(V)$ be its finite-dimensional complex irreducible representation. You can define two non-degenerate symmetric forms on $\mathfrak{g}$:
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Standard Killing form: $K(X,Y)=tr(ad_X\circ ad_Y)$
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"Killing-like" form associated with $\phi$: $K_{\phi}(X,Y)= tr(\phi(X)\phi(Y))$
In general, is there any connection between $K_\phi$ and $K$? I know for instance that for defining representation of $\mathfrak{gl}(N,\mathbb{C})$ both forms are proportional. Is it true for general semi-simple Lie algebra? If not, is there a separate name for $K_{\phi}$?
I'm asking this question because in math-physical literature connected to research I'm doing people tend to confuse these two forms: $K_\phi$ is used to define second order Casimir invariant of a given representation. Yet, in some articles there is simply $K$ instead of $K_{\phi}$.
Best Answer
They are proportional if $g$ is simple. The form $K_\phi$ defines a homomorphism from the adjoint to the coadjoint representation. If the adjoint representation is irreducible, i.e. $g$ is simple, you know all such homomorphisms are proportional by Schur's lemma.