Structure theorem for covariant bundles on quantum homogeneous spaces

The natural generalization of the notion of bundle in quantum geometry is that of bimodule. If the base space has quantum group symmetries, one is particularly interested in bimodules covariant (equivariant) under these symmetries. Most attention has so far been focused on the case with maximal symmetry — where the base space is a quantum group and the bimodules are bicovariant. The structure of bicovariant bimodules is well understood through their correspondence with crossed modules. We investigate the “next best” case — where the base space is a quantum homogeneous space and the bimodules are covariant. We present a structure theorem that resembles the one for bicovariant bimodules. Thus, there is a correspondence between covariant bimodules and a new kind of “crossed” modules which we define. The latter are attached to the pair of quantum groups which defines the quantum homogeneous space. We apply our structure theorem to differential calculi on quantum homogeneous spaces and discuss a related notion of induced differential calculus. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. This work was supported by a NATO fellowship grant.