In Chapter 70 of Srednicki's QFT he discusses what he calls the index of a representation T(R) defined by(adsbygoogle = window.adsbygoogle || []).push({});

Tr(T^{a}_{R}T^{b}_{R}) = T(R)δ^{ab}

I think other places call this a killing form, but I may be mistaken. In any case he discusses reducible representations R = R_{1}⊕R_{2}. He then states (eqn 70.11) that dim R = dim R_{1}+ dim R_{2}which is obvious. He then states (eqn 70.12) that T(R) = T(R_{1}) + T(R_{2}) which is what I want to know about. Is this (70.12) a result or an assertion? If it's a result, how do I see it? If it's an assertion, then why do we make this choice?

TIA

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# I Index (killing form ?) in a reducible representation

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