I haven't yet had time to dig deeper myself, but I think something is completely wrong with your assumption that the Cartan Matrix should be dependent on the Jz quantum number m. This cannot be true since the Cartan matrix is independent of any choice of representation. However, the allowed...
Spin 1 representation: In case of su(2), the spin algebra, there is exactly one irrep per dimension. The 1-dim irrep corresponds to heighest weight j=0, i.e. it describes a spin 0 system and is therefore called the spin 0 representation. The 2-dim irrep corresponds to spin 1/2, the 3-dim irrep...
Hey! I am studying group theory for particle physicists right now and I came across the following general question (Tell me if you think this rather belongs to the homework section, I am new here.)
Say I am given a set of matrices which represent the generators of an algebra, but I don't know...