The Heisenberg double of the quantum Euclidean group and its representations

The Heisenberg double D_(q)(E_(2))of the quantum Euclidean group O_(q)(E_(2))is the smash product of O_(q)(E_(2))with its Hopf dual U_(q)(e_(2)).For the algebra D_(q)(E_(2)),explicit descriptions of its prime,primitive and maximal spectra are obtained.All the prime factors of D_(q)(E_(2))are presented as generalized Weyl algebras.As a result,we obtain that the algebra D_(q)(E_(2))has no finite-dimensional representations,and D_(q)(E_(2))cannot have a Hopf algebra structure.The automorphism groups of the quantum Euclidean group and its Heisenberg double are determined.Some centralizers are explicitly described via generators and defining relations.This enables us to give a classification of simple weight modules and the so-called a-weight modules over the algebra D_(q)(E_(2)).