Most books that treat nonabelian gauge theory do not contain detailed discussion on Killing-Cartan forms, they'll usually just say that in $\text{SU}(N)$ Yang-Mills theory, one can choose generators $T_A$ such that $$ \text{Tr}(T_AT_B)=\frac{1}{2}\delta_{AB}, $$ implying that the inner product in the Lie algebra is $$ \langle T_A, T_B\rangle=2\text{Tr}(T_A T_B). $$

Even on the mathematical side, I mainly studied Lie groups from differential geometry texts that often completely ignored the Killing-Cartan form, or just barely mentioned it.

In short, I am looking for mainly physics-oriented texts that treat it in detail, including but not limited to

  • deriving necessary and sufficient conditions on the Lie group/algebra such that the Killing form is nondegenerate and definite (emphasis is on deriving, I can look up the conditions themselves in wikipedia)

  • various conventions and expressions relating it to the trace operation in matrix representations.

Whilst, as I have stated beforehand, I do not mind, in fact I even prefer physics-minded treatments, I don't even mind if the reference is working solely with matrix groups, I would like all proofs/derivations to be rigorous, rather than handwavy. I can get all facts from texts I already have, I am solely looking for proofs.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.