Quantum Heisenberg–Weyl Algebras

All Lie bialgebra structures on the Heisenberg–Weyl algebra [A+, A−] = M are classified and explicitly quantized. The complete list of quantum Heisenberg–Weyl algebras so obtained includes new multiparameter deformations, most of them being of the non-coboundary type. A Hopf algebra deformation of a universal enveloping algebra Ug defines in a unique way a Lie bialgebra structure (g, δ) on g [1]. The cocommutator δ provides the first order terms in the deformation of the coproduct, and can be seen as the natural tool to classify quantum algebras. Moreover, this well known statement suggests the relevance of the inverse problem, i.e., to find a method to construct, given an arbitrary Lie bialgebra, a Hopf algebra quantization of it. This question has been addressed recently in [2], where a very general construction of a deformed coassociative coproduct linked to a given Lie bialgebra (g, δ) has been presented. Such Lie bialgebra quantization formalism, inspired by the paper [3] (see also [4]), has been shown to be universal for the oscillator algebra: multiparametric coproducts corresponding to all coboundary oscillator Lie bialgebra structures can be obtained in that way (for the oscillator algebra non-coboundary structures do not exist [5]). To complete the structure of quantum algebras, deformed commutation rules can be found by imposing the homomorphism condition for the coproduct (counit and antipode can be also easily derived). In this letter we show that all Heisenberg–Weyl Lie bialgebras can be completely quantized by making use of this formalism. This result enhances the advantages of such an approach in order to obtain a full chart of Hopf algebra deformations of physically relevant algebras. Firstly, we find the most general form of all families of Heisenberg–Weyl Lie bialgebras. It is remarkable that, in contrast to the oscillator case, now there exists only one coboundary bialgebra among them. Afterwards, it is shown how all these Lie bialgebras can be classified and “exponentiated” to get the quantum coproducts by means of the formalism introduced in [2]. We also find all deformed commutation rules, thus obtaining a complete list of quantum deformations of this algebra, whose properties are briefly commented. This exhaustive description is fully complementary with respect to the quantum group results already known either from a Poisson–Lie construction [6] or from an R-matrix approach [7]. Let us fix the notation. The Heisenberg–Weyl Lie algebra h3 is generated by A+, A− and M with Lie brackets [A−, A+] = M, [M, · ] = 0. (1) A 3× 3 real matrix representation D of (1) is given by: