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Homogeneous spaces: From the classical to the quantum case. (English) Zbl 0952.16027

Szenthe, J. (ed.), New developments in differential geometry, Budapest 1996. Proceedings of the conference, Budapest, Hungary, July 27-30, 1996. Dordrecht: Kluwer Academic Publishers. 39-51 (1999).
The authors formulate the classical theory of homogeneous spaces of Lie groups in terms of algebras of functions on manifolds. This formulation is then extended to noncommutative algebras and Hopf algebras to define the notion of a quantum homogeneous space. The construction is illustrated by the quantum group \(\text{SU}_q(2)\) and the full family of quantum spheres of Podleś. The approach and the examples presented coincide with the ones of M. S. Dijkhuizen and T. H. Koornwinder [Geom. Dedicata 52, No. 3, 291-315 (1994; Zbl 0818.17017)] and T. Brzeziński [J. Math. Phys. 37, No. 5, 2388-2399 (1996; Zbl 0878.17011)].
For the entire collection see [Zbl 0903.00045].

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
22E99 Lie groups
46L87 Noncommutative differential geometry
58B32 Geometry of quantum groups
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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