NONCOMMUTATIVE PRINCIPAL BUNDLES 

 

The recognition of the domain of mathematics called fibre bundles took place in the period 1935 - 1940. The first general definitions were given by H. Withney. His work and that of H. Hopf and E. Stiefel demonstrated the importance of the subject for the applications of topology, differential geometry and theoretical physics. The question, whether there is a way to extend the classical theory of fibre bundles to noncommutative geometry is therefore of particular interest. In the case of vector bundles the Theorem of Serre and Swan gives the essential clue: The category of vector bundles over a compact space X is equivalent to the category of finitely generated and projective C(X)-modules. This observation leads to a notion of noncommutative vector bundles and is the connection between the topological K-theory based on vector bundles and the K-theory for C*-algebras. For principal bundles, free and proper actions of quantum groups on C*-algebras offer a good candidate for a notion of noncommutative principal bundles. In a purely algebraic setting, the well-established theory of Hopf-Galois extensions provides a wider framework comprising coactions of Hopf algebras.

 

The study and classification of actions of quantum groups on C*-algebras is intrinsically interesting and the experience with the commutative case suggests that free group actions are due to the lack of degeneracies easier to understand and to classify than general actions. In fact, by a classical result, having a free action of a compact group G on a compact space P is equivalent saying that P carries the structure of a principal bundle over the quotient X := P/G with structure group G. Very well-understood is the case of locally trivial principal bundles, that is, if P is glued together from spaces of the form U × G with an open subset U ⊆ X. This gluing immediately leads to G-valued cocycles. The corresponding cohomology theory, called Cech cohomology, gives a complete classification of locally trivial principal bundles with base space X and structure group G.

 

For compact Abelian groups, free ergodic actions, that is, free actions with trivial fixed point algebra, were completely classified by Olesen, Pedersen and Takesaki and independently by Albeverio and Høegh–Krohn. This classification was generalized to compact non-Abelian groups by the remarkable work of Wassermann: For a compact group G there is a 1-to-1 correspondence between free ergodic actions of G and 2-cocycles of the dual group. An analogous result in the context of compact quantum groups has been obtained by Bichon, De Rijdt and Vaes. Extending these classification results beyond the ergodic case is however not straightforward because, even for a commutative fixed point algebra, the action cannot necessarily be decomposed into a bundle of ergodic actions.

 

RESEARCH ACHIEVEMENTS

 

My research revolves around finding suitable algebraic analogs of useful concepts and ideas from fibre bundle theory and generalizing the corresponding classical results to the setting of noncommutative geometry. In particular, I am interested in the noncommutative geometry of principal bundles and their applications to geometry and physics.

 

My scientific work has brought an important insight into the theory of noncommutative principal bundles and my research results have been published as peer-reviewed articles in different journals, among which are Advances in Mathematics, the Proceedings of the London Mathematical Society and the Journal of Noncommutative Geometry.

 

For example, the articles [8, 9, 10, 11], which emerged from my PhD thesis, contain several aspects of the noncommutative geometry of principal bundles. Furthermore, Kay Schwieger and I achieved in [4] a complete classification of free, but not necessary ergodic actions of compact Abelian groups on unital C*-algebras. This classification relies on the fact that the corresponding isotypic components are Morita self-equivalences over the fixed point algebra.

 

For free actions of non-Abelian compact groups the bimodule structure of the corresponding isotypic components is more subtle. For this reason we concentrated in [5] on a simple class of free actions of non-Abelian compact groups, namely cleft actions which are characterized by the fact that all associated noncommutative vector bundles are trivial. Although this property looks limiting, in fact, many noncommutative phenomena already show up here.

 

In [3] we finally considered the general case of free actions of compact quantum groups on unital C*-algebras. To be more precise, we provided a complete classification of this class of actions in terms of generalized factor systems. Besides a new and interesting characterization of freeness, our approach uses the fact that non-ergodic actions of compact quantum groups can be described in terms of weak unitary tensor functors mapping the representation category of the underlying compact quantum group into the category of C*-correspondences over the corresponding fixed point algebra.

 

As an application of our generalized factor system theory, we showed that finite noncommutative coverings of generic irrational rotation C*-algebras are always cleft. Recently, we have finished an article [1] that aims at understanding noncommutative coverings of quantum tori.

 

CURRENT RESEARCH DESCRIPTION

 

My current research plan is divided into two major parts, which both are geometric in nature and require a deep understanding of noncommutative principal bundles. In fact, the overall purpose of the first part is to study the geometry of noncommutative principal bundles by means of Connes’ spectral triples, which provide a natural framework to do geometry by analogy with the Riemannian spin geometry of a classical manifold. The specific objective of the second part is to explore strategies towards a fundamental group for C*-algebras. In particular, I hope to find interesting and important invariants of C*-algebras complementing the recent extensive activity with the attempt to extend Elliott’s classification program for C*-algebras.