Representations of Lie algebras from representations of quantum groups

In paper [*] (P. Moylan: Czech. J. Phys., Vol. 47 (1997), p. 1251) we gave an explicit embedding of the three dimensional Euclidean algebra ε(2) into a quantum structure associated with U_q(so(2,1)). We used this embedding to construct skew symmetric representations of ε(2) out of skew symmetric representations of U_q(so(2, 1)). Here we consider generalizations of the results in [*] to a more complicated quantum group, which is of importance to physics. We consider U_q(so(3, 2)), and we show that, for a particular representation, namely the "Rac" representation, many of the results in [*] carry over to this case. In particular, we construct representations of so(3, 2), P(2, 2), the Poincaré algebra in 2+2 dimensions, and the Poincaré algebra out of the Rac representation of U_q(So(3, 2)). These results may be of interest to those working on exploiting representations of U_q(So(3, 2)), like the Rac, as an example of kinematical confinement for particle constituents such as the quarks.