Reduction of a Class of O(4, 2) Representations with Respect to SO(4, 1) and SO(3, 2)

A complete classification of representations of SO(4, 2) with infinitesimal generators S_AB, characterized by the representation relation {S_AB, S^A_C} = −2ag_BC, and their extension by parity have been determined. The possible values of a are a = 1 − S², $\text{S = 0,}\frac{1}{2}\text{, 1,}\frac{3}{2}\text{, 2, ⋯.}$ These representations have then been reduced according to the two chains $\text{SO(4,2)⊃SO(4,1)⊃SO(4)⊃SO(3)}$ and $\text{SO(4,2)⊃SO(3,2)⊃SO(3)⊗SO(2).}$ The equivalence of these representations with the oscillator representations is established.