REPRESENTATIONS OF LIE ALGEBRAS FROM REPRESENTATIONS OF QUANTUM GROUPS

Abstract

In paper [*] (P. Moylan: Czech. J. Phys., Vol. 47 (1997), p. 1251) we gave an explicit embedding of the three dimensional Euclidean algebra ɛ(2) into a quantum structure associated with Uq(so(2, 1)). We used this embedding to construct skew symmetric representations of ɛ(2) out of skew symmetric representations of Uq(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 Uq(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 Poincare algebra in 2+2 dimensions, and the Poincare algebra out of the Rac representation of Uq(so(3, 2)). These results may be of interest to those working on exploiting representations of Uq(so(3, 2)), like the Rac, as an example of kinematical confinement for particle constituents such as the quarks.