Quantum Homogeneous Spaces, Duality, and Quantum 2-Spheres

摘要

For a quantum group G, the notion of quantum homogeneous G-space is defined. Twomethods to construct such spaces are discussed. The first one makes use of quantum subgroups, the second more general one is based upon the notion of infinitesimal invariance with respect to certain two-sided coideals in the Hopf algebra dual to the Hopf algebra of G. These methods are applied to the quantum group SU(2). As two-sided coideals the authors take the subspaces spanned by twisted primitive elements in the Sl(2) quantized universal enveloping algebra. A one-parameter series of mutually non-isomorphic quantum 2-spheres is obtained, together with the spectral decomposition of the corresponding right regular representation of quantum SU(2). The link with the quantum spheres defined by Podles is established.