Understanding Equation XI.31 in Lie Algebras

[vega12]

I am currently trying to up my understanding of Lie algebras as the brief introductions I have had from various QFT textbooks feels insufficient, but have been stuck on one small point for a couple days now. I am reading through the lecture notes / book by Robert Cahn found here: http://theory.uchicago.edu/~sethi/Teaching/P385-W2011/texall.pdf. On page 97, I am having trouble understanding how equation XI.31 comes about. I think I get how Tr C = N_\Lambda \langle \Lambda, \Lambda + 2 \delta \rangle_2 comes about, but don't see how I can use the statement regarding (( , )) yielding l_\phi ( , )_2 and how that directly leads to XI.31.

A bit of guidance here would be very much appreciated. Thanks.

 

[vega12]

I am very sorry for bumping my own thread with a double post. If it turns out that I cannot get a response here, could someone perhaps recommend a more appropriate subforum for me to post in? Would "general math" be more promising? Thanks.

 

[chiro]

Hey vega12 and welcome to the forums.

With the normalization procedure, what is the definition of something with unit length in the context of the operator you are working with (the matrix)?