Noncommutative Instantons on the 4-Sphere¶from Quantum Groups

We describe an approach to the noncommutative instantons on the 4-sphere based on quantum group theory. We quantize the Hopf bundle ?7→?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 SUq(2); it determines a new deformation of the 4-sphere ∑4q as the algebra of coinvariants in ?q7. We show that the quantum vector bundle associated to the fundamental corepresentation of SUq(2) is finitely generated and projective and we compute the explicit projector. We give the unitary representations of ∑4q, 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.