Noncommutative Instantons on the 4-Sphere from Quantum Groups

Abstract

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 Sigma^4_q as the algebra of coinvariants in S_q^7. 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 Sigma^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.