Poisson-Hopf limit of quantum algebras

TL;DR

This paper introduces a Poisson-Hopf analogue of quantum algebras by constructing a limit of quantizations depending on a parameter h, extending q-deformations to the classical Poisson setting, with potential applications in semiclassical physics.

Contribution

It develops a systematic method to derive Poisson-Hopf algebras from quantum algebras, including new cases like su_q^P(3), and extends q-Serre relations to the Poisson limit.

Findings

Constructed Poisson-Hopf algebras from quantum algebras.

Introduced the su_q^P(3) algebra as a novel case.

Extended q-Serre relations to the Poisson framework.

Abstract

The Poisson-Hopf analogue of an arbitrary quantum algebra U_z(g) is constructed by introducing a one-parameter family of quantizations U_{z,h}(g) depending explicitly on h and by taking the appropriate h -> 0 limit. The q-Poisson analogues of the su(2) algebra are discussed and the novel su_q^P (3) case is introduced. The q-Serre relations are also extended to the Poisson limit. This approach opens the perspective for possible applications of higher rank q-deformed Hopf algebras in semiclassical contexts.