Abstract
The formalism of a classical homogeneous spase is encoded by using the language of Hopf algebras. In this way a natural definition of a quantum homogeneous space is presented. We study the simplest case of a quantum 2-sphere, and investigate its tangent space via pseudo-vector fields. A quantum analogue of the polar decomposition for the coordinates of this quantum sphere is also investigated.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E. Abe: Hopf Algebras, Cambridge Univ. Press, 1980.
A. Arvanitoyeorgos and D. Ellinas: Quantum (co)modules for dual quantum groups, Submitted for publication.
G. E. Andrews: q-Series: Their Development and Applications in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra, AMS CBMS 1986.
F. Bonechi, N. Ciccoli, R. Giachetti, E. Sorace and M. Tarlini: Invariant q-Schrödinger Equation from homogeneous spaces of the 2-dim Euclidean quantum group, Comm. Math. Physics 175 (1996) 161.
C-S. Chu, P-M. Ho and B. Zumino: The quantum 2-sphere as a complex manifold, (preprint 1995, LBL-37055, UCB-PTH-95/10, Z. Phys. C., to appear).
V. G. Drinfel’d: Quantum groups, Proc. ICM Berkeley CA, 1987, Ed. A. M. Gleason, AMS, Providence RI.
M. S. Dijkhuizen and T. H. Koornwinder: Quantum homogeneous spaces, duality and quantum 2-spheres, Geom. Dedicata 52 (1994) 291–315.
D. Ellinas: Path integrals for quantum algebras and the classical limit, J. Phys. A Gen. Math. A26 (1993) L543.
D. Ellinas: Phase operators via group contraction, Jour. Math. Phys. 32, 135 (1991).
L. D. Faddeev, N. Yu Reshetikhin and L. A. Takhtadjan: Quantization of Lie groups and Lie algebras, Leningrand Math. J. 1, 193 (1990).
P. R. Halmos: A Hilbert Space Problem Book, Springer Berlin 1982.
Y. I. Manin: Quantum groups and noncommutative geometry, Centre de Recherches Mathématiques 1561, Université de Montréal (1988).
S. Montgomery: Hopf Algebras and Their Actions on Rings, Amer. Math. Society CBMS (82) (1993).
M. Noumi, K. Mimachi: A skey-Wilson polynomials as spherical functions on SU q(2), in: P. P. Kulish ed. Quantum Groups LNM 1510, Spinger Verlag, New York, 1992 98–103.
A. M. Perelomov: Generalized Coherent States and their Applications, Springer Verlag, Berlin 1985.
P. Podles: Quantum Spheres, Lett. Math. Phys. 14, 193 (1987).
M. Reed, B. Simon: Methods in Modern Mathematical Physics I, Functional Analysis, Academic, New York 1972. p.139.
M. Sweedler: Hopf Algebras, W. A. Benjamin, 1969.
L. L. Vaksman and Y. S. Soibel’man: Algebra of functions on the quantum groups SU(2), Func. Anal. Appl. 22, 170 (1988).
F. W. Warner: Foundations of Differential Manifolds and Lie Groups, Scott, Foresman, Glenview, Illinois, 1971.
S. L. Woronowicz: Twisted SU(2) group. An example of noncommutative differential calculus, Publ. RISM, Kyoto Univ. 23, 117 (1987).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Arvanitoyeorgos, A., Ellinas, D. (1999). Homogeneous Spaces: From the Classical to the Quantum Case. In: Szenthe, J. (eds) New Developments in Differential Geometry, Budapest 1996. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5276-1_2
Download citation
DOI: https://doi.org/10.1007/978-94-011-5276-1_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6220-6
Online ISBN: 978-94-011-5276-1
eBook Packages: Springer Book Archive