  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.
