I Index (killing form ?) in a reducible representation

[hideelo]

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

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 = 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

 

[fresh_42]

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

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 = 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.