Quantum algebras as quantizations of dual Poisson-Lie groups

Abstract

A systematic computational approach for the explicit construction of any quantum Hopf algebra (U_z(g),\Delta_z) starting from the Lie bialgebra (g,\delta) that gives the first-order deformation of the coproduct map \Delta_z is presented. The procedure is based on the fact that any quantum algebra can be viewed as the quantization of the unique Poisson-Lie structure (G^\ast,\Lambda_g) on the dual group G^\ast, which is obtained by exponentiating the Lie algebra g^\ast defined by the dual map \delta^\ast. From this perspective, the coproduct for U_z(g) is just the pullback of the group law for G^\ast, and the Poisson analogues of the quantum commutation rules for U_z(g) are given by the unique Poisson-Lie structure \Lambda_g on G^\ast whose linearization is the Poisson analogue of the initial Lie algebra g. This approach is shown to be very useful in order to construct quantum deformations explicitly since, once a Lie bialgebra (g,\delta) is given, the full dual Poisson-Lie group (G^\ast,\Lambda) can be obtained either by applying standard Poisson-Lie group techniques or by implementing the algorithm here presented with the aid of symbolic manipulation programs. As a consequence, the quantization of (G^\ast,\Lambda) will give rise to the full U_z(g) quantum algebra, provided that ordering problems are appropriately fixed. The applicability of this approach is explicitly demonstrated by constructing several instances of quantum deformations of physically relevant Lie algebras as sl(2,R), the (2+1) Anti de Sitter algebra so(2,2) and the Poincar\'e algebra in (3+1) dimensions.