Quantum phase space of the group of affine transformations of the line

The problem of constructing phase spaces to quantum groups and algebras is a mystifying one even in the quasiclassical limit, due to lack of general principles. We consider the simplest situation of the group G of affine transformations of the line. G has a two-dimensional space of Poisson-Lie structures. For each such fixed structure, the space of Poisson structures on the cotangent space T*G is infinite-dimensional, but there exists a unique one which is both left- and right-invariant. All this is quantized, including the dual space G*, to the Lie algebra G of G and the coadjoint representation of G on G*. (orig.)