Quantum algebras as quantizations of dual Poisson–Lie groups

A systematic computational approach for the explicit construction of any quantum Hopf algebra (Uz(g), Δz) starting from the Lie bialgebra (g, δ) that gives the first-order deformation of the coproduct map Δz is presented. The procedure is based on the well-known 'quantum duality principle', namely the fact that any quantum algebra can be viewed as the quantization of the unique Poisson–Lie structure (G*, Λg) on the dual group G*, which is obtained by exponentiating the Lie algebra g* defined by the dual map δ*. From this perspective, the coproduct for Uz(g) is just the pull-back of the group law for G*, and the Poisson analogues of the quantum commutation rules for Uz(g) are given by the unique Poisson–Lie structure Λg on G* whose linearization is the Poisson analogue of the initial Lie algebra g. This approach is shown to be a very useful technical tool in order to solve the Lie bialgebra quantization problem explicitly since, once a Lie bialgebra (g, δ) is given, the full dual Poisson–Lie group (G*, Λ) can be obtained either by applying standard Poisson–Lie group techniques or by implementing the algorithm presented here with the aid of symbolic manipulation programs. As a consequence, the quantization of (G*, Λ) will give rise to the full Uz(g) quantum algebra, provided that ordering problems are appropriately fixed through the choice of certain local coordinates on G* whose coproduct fulfils a precise 'quantum symmetry' property. The applicability of this approach is explicitly demonstrated by reviewing the construction of several instances of quantum deformations of physically relevant Lie algebras such as sl(2,R), the (2+1) anti-de Sitter algebra so(2, 2) and the Poincaré algebra in (3+1) dimensions. (paper)