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
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