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

The Chern-Galois character

Following the idea of Galois-type extensions and entwining structures, we define the notion of a principal extension of noncommutative algebras. We show that modules associated to such extensions via finite-dimensional corepresentations are finitely generated projective, and determine an explicit formula for the Chern character applied to the thus obtained modules.Comment: 4 pages, LaTe

Nontrivial Deformation of a Trivial Bundle

The ${\rm SU}(2)$-prolongation of the Hopf fibration $S^3\to S^2$ is a trivializable principal ${\rm SU}(2)$-bundle. We present a noncommutative deformation of this bundle to a quantum principal ${\rm SU}_q(2)$-bundle that is not trivializable. On the other hand, we show that the ${\rm SU}_q(2)$-bundle is piecewise trivializable with respect to the closed covering of $S^2$ by two hemispheres intersecting at the equator.Comment: The present paper has been extracted from an earlier version of arXiv:1101.0201, so that there are some overlaps in introductory parts and standard definition