The Heisenberg double of the quantum Euclidean group and its representations

The Heisenberg double Dq(E2) of the quantum Euclidean group $${{\cal O}_q}({E_2})$$ is smash product with its Hopf dual $${U_q}({e_2})$$ . For algebra Dq(E2), explicit descriptions prime, primitive and maximal spectra are obtained. All prime factors presented as generalized Weyl algebras. As a result, we obtain that has no finite-dimensional representations, cannot have structure. automorphism groups determined. Some centralizers explicitly described via generators defining relations. This enables us to give classification simple weight modules so-called a-weight over Dq(E2).