Lie Algebra

The commutation relation of Lie Algebra 

with the specific commutation relations defined by the structure constants, completely determine the entire structure of the group.

One particular representation formed by the structure constant themselves:

 where 

is called the Adjoint Representation:  number of  matrices.

The  representation of  is  dimensional with eigenvalues of  are

Particles with spin  are described by the  representation of .

For the  representation of , there are three  generators , , and (the number of generators is determined by the number of continuous parameters, which is determined by the group itself and has nothing to do with which representation we choose ). By construction  is diagonal with eigenvalues