Quantum E(2) group and its Pontryagin dual

The quantum deformation of the group of motions of the plane and its Pontryagin dual are described in detail. It is shown that the Pontryagin dual is a quantum deformation of the group of transformations of the plane generated by translations and dilations. An explicit expression for the unitary bicharacter describing the Pontryagin duality is found. The Heisenberg commutation relations are written down.Supported in equal parts by the grant of the Ministry of Education of Poland and by Schweizerischer Nationalfonds.