Dirac operator on quantum homogeneous spaces and noncommutative geometry

Spectral triples constitute basic structures of Connes' noncommutative geometry. In order to unify noncommutative geometry with quantum group theory, it is necessary to provide a proper description of spectral triples on quantum groups and, in a wider context, on their homogeneous spaces. Thus, a way to define Dirac-like operators on such quantum spaces has to be found. This paper is a brief summary on some problems we are facing in searching for such a way. We suggest that some modifications of either noncommutative geometry or quantum group theory, or both, are inevitable. Differential calculi and their relationship to the Dirac operators are the key concepts to understand for defining the spectral triples in question. Since there is no canonical way to construct such calculi on the quantum groups, and since there are two basic kinds, left-(right-) covariant and bicovariant, there are a number of issues to be resolved. We discuss these difficulties as well as difficulties implicit in the definition of the Dirac operator, which are present already at the classical manifolds level. Definition of quantum homogeneous spaces constitutes another problem. This is also briefly reviewed in the paper. Then a basic example associated with the quantum 2-sphere of Podles, as a homogeneous space of the quantum SU(2) group of Woronowicz, is discussed. A naturally defined Dirac operator on such a 2-sphere does not satisfy one of Connes' axioms.