>>14188386 #I am not a type of coffee
>>14188711 #First, download "Lie algebras in particke physics" by Howard Georgi for the chapters on su2, su3 and roots.
Ok, so yes L^2 is a also bit random to me too. Its allegedly connected to a "casimir element" which i dont know much about and it was studied by mathematicians way before quantum mechanics was a thing. There is an alternative formalism which you know. its where your operators are not Lx, Ly, Lz or L^2 but just Lz, L+,L-, where L+=Lx+iLy, etc. The famous raising and lowering operators. This treatment allows you do several things:
-Put an algebra into an international standard form
-Get a canonical base [n,l,m> for eigenvectors. Get all the eigenvalues.
-An algorithm to get the actual matrix representations of the algebra. For su(2) there are representations for any n equal or above 2. Each of these representations are labeled by the index "l" of fame as a quantum number in QM. I think in math this l number is called the "highest weight" and it means the highest positive eigenvalue of a representation, which in a way also works to label the representation. In su(2) theres a representation for l=1/2,1,3/2 and so on but theres also nearly identical treatment for any simple complex lie algebra (hehe simple complex).
These algebras include all algebras classified by something called Dynkin diagrams, which includes su(n) so(2n+1 ), symplectic algebra, so(2n) and autistic special algebras like f4, g2, e6, e7, e8 which are often mentioned in popsi. Eli Cartan and his homies worked this out in like 1908. In general a random algebra like su(n) doesnt have just one element like Lz but it has what is called a Cartan Subalgebra, which in su(2) only has 1 element, Lz.
