  The Heisenberg double of a Hopf algebra may be regarded as a quantum analogue
of the cotangent bundle of a Lie group. Quantum duality principle describes
relations between a Hopf algebra, its dual, and their Heisenberg double in a
way which extends both the theory of coadjoint orbits and the classical Fourier
transform. We also describe the twisted Heisenberg double which is relevant for
the study of nontrivial deformations of the quantized universal enveloping
algebras.
