The three-dimensional Euclidean quantum group E(3)q and its R-matrix

A contraction procedure starting from SO(4)q is used to determine the quantum analog E(3)q of the three-dimensional Euclidean group and the structure of its representations. A detailed analysis of the contraction of the R-matrix is then performed and its explicit expression has been found. The classical limit of R is shown to produce an integrable dynamical system. By means of the R-matrix the pseudogroup of the noncommutative representative functions is considered. It will finally be shown that a further contraction made on E(3)q produces the two-dimensional Galilei quantum group and this, in turn, can be used to give a new realization of E(3)q and E(2,1)q.