Table of Contents

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

Quantum Frame Bundles

A very interesting class of quantum principal bundles is given by frame bundles. They provide a nice working framework to incorporate various geometrical structures (including Riemannian, symplectic, complex and spin structures) into the quantum context. The main idea is to define quantum frame bundles axiomatically, as in the classical geometry [KN], starting from the idea of cannonical coordinate first-order horizontal forms. In classical geometry, the algebra of horizontal forms is realizable as the tensor product of the functions on the bundle with the exterior algebra associated to the space of coordinate first-order forms. This property can be used [D10] as a starting point in defining quantum frame bundles-in quantum case the space of horizontal forms is defined as where is a vector space defining quantum first-order horizontal forms, and is a braided exterior algebra built over , relative to the appropriate braid operator . This braid operator comes from the theory of bicovariant bimodules [W3]--because it is assumed that is a left-invariant part of a bicovariant *-bimodule over G. This also implies that G is naturally acting by a *-homomorphism Taking a product of the actions F and we can define the structure quantum group action

It is important to mention that the algebra structure on hor(P) is not a simple tensor product, but the appropriate cross-product of and so that above mentioned structure group coaction is a *-homomorphism. To complete the geometrical picture of a quantum frame bundle, it is necessary to postulate the existence of the appropriate differential calculus on the *-algebra of G-invariant elements of hor(P).

Quantum frame bundles also provide a class of examples of principal bundles where it is possible to built an intrinsic differential caclulus on the bundle, applying a general constructive approach to differential calculi, developed in [D6]. In the framework of such a calculus, it is possible to incorporate into the quantum context the entire formalism of torsion operators, and to generalize various important constructions of clasical Riemannian/spin geometry--including the study of Levi-Civita connections and constructions of a quantum Dirac operator [D10][D11].