On a Morita equivalence between the duals of quantum SU(2) and quantum E˜(2)

Let SUq(2) and E˜q(2) be Woronowicz's q-deformations of respectively the compact Lie group SU(2) and the non-trivial double cover of the Lie group E(2) of Euclidean transformations of the plane. We prove that, in some sense, their duals are 'Morita equivalent locally compact quantum groups'. In more concrete terms, we prove that the von Neumann algebraic quantum groups L∞(SUq(2)) and L∞(E˜q(2)) are unitary cocycle deformations of each other.