# Index (killing form ?) in a reducible representation

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## Main Question or Discussion Point

In Chapter 70 of Srednicki's QFT he discusses what he calls the index of a representation T(R) defined by

Tr(TaR TbR) = T(R)δab

I think other places call this a killing form, but I may be mistaken. In any case he discusses reducible representations R = R1⊕R2. He then states (eqn 70.11) that dim R = dim R1 + dim R2 which is obvious. He then states (eqn 70.12) that T(R) = T(R1) + T(R2) 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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In Chapter 70 of Srednicki's QFT he discusses what he calls the index of a representation T(R) defined by

Tr(TaR TbR) = T(R)δab

I think other places call this a killing form, but I may be mistaken. In any case he discusses reducible representations R = R1⊕R2. He then states (eqn 70.11) that dim R = dim R1 + dim R2 which is obvious. He then states (eqn 70.12) that T(R) = T(R1) + T(R2) 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
If you have a direct sum of vector spaces, on which something operates in a way, that leaves both subspaces ##R_i## invariant, then this operation can be written in block matrix form ##T_R= \begin{bmatrix}T_{R_1}&0\\0&T_{R_2}\end{bmatrix}## and the traces can be added.