Quantum algebras as quantizations of dual Poisson–Lie groups

Abstract

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 , the (2+1) anti-de Sitter algebra so(2, 2) and the Poincaré algebra in (3+1) dimensions.