We describe an approach to the noncommutative instantons on the 4-sphere based on quantum group theory. We quantize the Hopf bundle S 7 → S 4 making use of the concept of quantum coisotropic subgroups. The analysis of the semiclassical Poisson-Lie structure of U (4) shows that the diagonal SU (2) must be conjugated to be properly quantized. The quantum coisotropic subgroup we obtain is the standard SU q (2); it determines a new deformation of the 4-sphere Σ 4 q as the algebra of coinvariants in S 7 q . We show that the quantum vector bundle associated to the fundamental corepresentation of SU q (2) is finitely generated and projective and we compute the explicit projector. We give the unitary representations of Σ 4 q , we define two 0-summable Fredholm modules and we compute the Chern-Connes pairing between the projector and their characters. It comes out that even the zero class in cyclic homology is non trivial.

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.

In this letter we derive a deformed Dirac equation invariant under the κ-Poincaré quantum algebra. A peculiar feature is that the square of the κ-Dirac operator is related to the second Casimir (the κ-deformed squared Pauli-Lubanski vector). The "spinorial" realization of the κ-Poincaré is obtained by a contraction of the coproduct of the real form of SO q (3, 2) using the 4-dimensional representation which results to be, up some scalar factors, the same of the undeformed algebra in terms of the usual γ-matrices.

The structure of the quantum Heisenberg group is studied in the two different frameworks of the Lie algebra deformations and of the quantum matrix pseudogroups. The R-matrix connecting the two approaches, together with its classical limit r, are explicitly calculated by using the contraction technique and the problems connected with the limiting procedure discussed. Some unusual properties of the quantum enveloping Heisenberg algebra are shown.