Quantum principal bundles over quantum real projective spaces

Two hierarchies of quantum principal bundles over quantum real projective spaces are constructed. One hierarchy contains bundles with U(1) as a structure group, the other has the quantum group $SU_q(2)$ as a fibre. Both hierarchies are obtained by the process of prolongation from bundles with the cyclic group of order 2 as a fibre. The triviality or otherwise of these bundles is determined by using a general criterion for a prolongation of a comodule algebra to be a cleft Hopf-Galois extension.Comment: 15 pages; v2 typos and omissions corrected, a discussion of Fredholm modules adde

The Einstein Action for Algebras of Matrix Valued Functions - Toy Models

Two toy models are considered within the framework of noncommutative differential geometry. In the first one, the Einstein action of the Levi-Civita connection is computed for the algebra of matrix valued functions on a torus. It is shown that, assuming some constraints on the metric, this action splits into a classical-like, a quantum-like and a mixed term. In the second model, an analogue of the Palatini method of variation is applied to obtain critical points of the Einstein action functional for M\sb 4(R). It is pointed out that a solution to the Palatini variational problem is not necessarily a Levi-Civita connection. In this model, no additional assumptions regarding metrics are made.Comment: 9 pages, AMS-LaTeX, serious typesetting problems due to 2.09-2.e incompatibility removed, reference adde

Projective module description of the q-monopole

The Dirac q-monopole connection is used to compute projector matrices of quantum Hopf line bundles for arbitrary winding number. The Chern-Connes pairing of cyclic cohomology and K-theory is computed for the winding number -1. The non-triviality of this pairing is used to conclude that the quantum principal Hopf fibration is non-cleft. Among general results, we provide a left-right symmetric characterization of the canonical strong connections on quantum principal homogeneous spaces with an injective antipode. We also provide for arbitrary strong connections on algebraic quantum principal bundles (Hopf-Galois extensions) their associated covariant derivatives on projective modules.Comment: AMS-LaTeX 18 pages, no figures, correction of the Chern-number-sign-change Comments, 6 pages of new contents adde