Hi all, I asked this on the Quantum Physics board but didn't get a response. I'm reading Cahn's book on semi-simple lie algebras and their representations. http://www-physics.lbl.gov/~rncahn/book.html In chapter 1, he attempts to build a (2j+1)-dimensional representation [itex]T[/itex] of the Lie algebra of SO(3) starting with the abstract commutation relations [itex][T_z,T_+] = T_+, \quad [T_z,T_-] = - T_-,\quad [T_+,T_-] = 2T_z[/itex] Eq (I.14). He begins by defining the action of [itex]T_z,T_+[/itex] on the vector [itex]v_j[/itex] by [itex]T_z v_j = j v_j, \quad T_+ v_j = 0[/itex] but he does not explain what the [itex]v_j[/itex]'s are. How does one even know that such vectors exist? Any help would be greatly appreciated.