Introduction

The homogeneous spaces of the Lie groups of the classical kinematics symmetries are between the most relevant topics in theorethical and mathematical physics. Indeed the equations generated by the action of the algebra on such spaces give rise to the fundamental equations of the classical physics and wave mechanics. Therefore it is a scientific meaningful program to study the analogous structures in the quantum groups framework.

The first step is the building of the quantum counterpart of the noncompact Lie groups. This task is now satisfactorily accomplished, at present we know explicitely the quantum versions – recovered by means of various methods – of all the relevant inhomogeneous Lie groups: from Heisenberg and Galilei to Euclides and Poincaré, (1+1)-dimensional and (3+l)-dimensional.

The second step is connected to the definition of the quantum homogeneous spaces, once a quantum group is given. Indeed, owing to the non commutativity of the group parameters, the quantum manifolds do not exist. However we can try to deal with manifolds in the quantum world by translating definitorial relations from the spaces to the functions on them and by using the duality between the “functions on the group” and the envelopping algebra. Therefore a crucial point is the injection of the algebra of the quantum functions on the homogeneous space into the algebra of the quantum functions on the group. The methodical approach is to express the classical construction of the homogeneous spaces in the language of Hopf algebras, that is in a way independent from the commutativity of the functions on the group.