CLASSICAL DEFORMATIONS, POISSON-LIE CONTRACTIONS, AND QUANTIZATION OF DUAL LIE BIALGEBRAS

A Poisson-Hopf algebra of smooth functions is simultaneously constructed on the two dimensional Euclidean, Poincaré, and Heisenberg groups by using a classical r-matrix which is invariant under contraction. The quantization for this algebra of functions is developed, and its dual Hopf algebra is also computed. Contractions on these quantum groups are studied. It is shown that, within this setting, classical deformations are transformed into quantum ones by Hopf algebra duality and the quantum Heisenberg algebra is derived by means of a (dual) Poisson-Lie quanti-zation that deforms the standard Moyal-Weyl *ℏ-product. © 1995 American Institute of Physics.