Table of Contents

Introduction
Differential Calculus
Connections Formalism
Characteristic Classes
Quantum Classifying Spaces
Quantum Frame Bundles
Associated Bundles
Gauge Transformations
Literature

Associated Vector Bundles

Starting from a quantum principal bundle P it is possible to define the concept of the associated vector bundle, essentially the same way as in the classical theory. In the framework of the formalism, the associated bundles appear as certain bimodules over . More precisely, to every representation u of G in a finite-dimensional vector space , we can associate a -bimodule consisiting of all intertwiners between the representation and the action F. In other words, the maps f are such that the diagram

is commutative. The elements of play the role of smooth sections of the actual associated vector bundle.

It turns out that the bimodules are finite and projective on both sides. The association preserves the direct sums, products and conjugation operation in the category of representations of G. In other words, we have

Here, the bars denote the conjugate bimodule and the representation. It is important to point out a difference between this definition of vector bundles, and the definition of vector bundles as finite projective one-sided modules [C] over the base space algebra. Our definition is intrinsically connected with the idea of a quantum principal bundle. Indeed, it can be shown that the system of all bimodules contains the complete information about the initial quantum principal bundle P, and accordingly it is possible to reconstruct the bundle starting from the system of associated bundles [D5]. This construction is based on the quantum version [W4] of classical Tannaka-Krein duality theory.

Cohomological invariants of the bundle are naturally connected with the system of associated bundles. Accordingly, it is possible to introduce the appropriate K-theory and the Chern character. These entities differ from the standard K-theory and the Chern character, defined in the framework of cyclic cohomology theory [C][K]. However both constructions can be inccorporated into the unifying framework of KK-theory [JT].