Abstract
We propose a formulation of the quantization problem of Manin quadruples, and show that a solution to this problem yields a quantization of the corresponding Poisson homogeneous spaces. We then solve both quantization problems in an example related to quantum spheres.
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
Andruskiewitsch, N.: Some exceptional compact matrix pseudogroups, Bull. Soc. Math. France 120 (1992), 297–325.
Bonneau, P., Flato, M.,C Gerstenhaber, and M. Pinczon, G.: The hidden group structure of quantum groups: strong duality, rigidity, and preferred deformations, Comm. Math. Phys. 161 (1994), 125–156.
Dazord, P. and Sondaz, D.: Groupes de Poisson affines, in: P. Dazord and A. Weinsten, (eds.), Symplectic geometry, groupoids, and integrable systems (Berkeley 1989), Math. Sci. Res. Inst. Publ., 20, Springer-Verlag, New York, 1991.
Delorme, P.: Sur les triples de Manin pour les algèbres réductives complexes, preprint math/9912055.
Donin, J., Gurevich, D., and Shnider, S.: Double quantization on some orbits in the coadjoint representations of simple Lie groups, Comm. Math. Phys. 204 (1999), 39–60.
Drinfeld, V: Quantum groups, in: Proceedings of the International Congress of Mathematicians (Berkeley 1986), Amer. Math. Soc, Providence, RI, 1987, pp. 798–820.
Drinfeld, V.: Quasi-Hopf algebras, Leningrad Math. J. 1 (1990), 1419–1457.
Drinfeld, V.: On Poisson homogeneous spaces of Poisson-Lie groups, Theoret. and Math. Phys. 95(1993), 524–525.
Enriquez, B. and Rubtsov, V.: Quasi-Hopf algebras associated with 1d5981d5912 complex curves, Israel J. Math. 112 (1999), 61–108.
Enriquez, B. and Felder, G.: Commuting differential and difference operators associated with complex curves II, preprint math/9812152.
Etingof, P. and Kazhdan, D.: Quantization of Poisson algebraic groups and Poisson homogeneous spaces, in: A. Connes, K. Gawedzki, and J. Zinn-Justin, (eds.), Symétries quantiques (Les Houches 1995), North-Holland, Amsterdam, 1998, pp. 935–946,
Karolinsky, E.: A classification of Poisson homogeneous spaces of compact Poisson-Lie groups, Dokl. Math., 57 (1998), 179–181. See also math/9901073.
Lu, J.-H.: Multiplicative and affine Poisson structures on Lie groups, Ph.D. thesis, Univ. of Berkeley, 1990.
Lu, J.-H.: Classical dynamical r-matrices and homogeneous Poisson structures on G/H and K/T, preprint math/9909004.
Noumi, M.: Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Adv. Math. 123 (1996), 16–77.
Parmentier, S.: On coproducts of quasi-triangular Hopf algebras, St. Petersburg Math. J. 6 (1995), 879–894.
Podles, P.: Quantum spheres, Lett. Math. Phys. 14 (1987), 193–202.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Kluwer Academic Publishers
About this chapter
Cite this chapter
Enriquez, B., Kosmann-Schwarzbach, Y. (2000). Quantum homogeneous spaces and quasi-Hopf algebras. In: Dito, G., Sternheimer, D. (eds) Conférence Moshé Flato 1999. Mathematical Physics Studies, vol 21/22. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-1276-3_8
Download citation
DOI: https://doi.org/10.1007/978-94-015-1276-3_8
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5551-4
Online ISBN: 978-94-015-1276-3
eBook Packages: Springer Book Archive