The commutation relation of Lie Algebra
![\left[X_{i}, X_{j}\right]=i f_{i j k} X_{k}](https://math.jianshu.com/math?formula=%5Cleft%5BX_%7Bi%7D%2C%20X_%7Bj%7D%5Cright%5D%3Di%20f_%7Bi%20j%20k%7D%20X_%7Bk%7D)
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:
![\left[T^{a}, T^{b}\right]=i f_{a b c} T^{c}](https://math.jianshu.com/math?formula=%5Cleft%5BT%5E%7Ba%7D%2C%20T%5E%7Bb%7D%5Cright%5D%3Di%20f_%7Ba%20b%20c%7D%20T%5E%7Bc%7D)
where ![\left[T^{a}\right]_{b c}=-i f_{a b c}](https://math.jianshu.com/math?formula=%5Cleft%5BT%5E%7Ba%7D%5Cright%5D_%7Bb%20c%7D%3D-i%20f_%7Ba%20b%20c%7D)
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 %0A)
, there are three *(2j%2B1))
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
