Abstract
We study the quantum sphere as a quantum Riemannian manifold in the quantum frame bundle approach. We exhibit its 2-dimensional cotangent bundle as a direct sum Ω0,1⊕Ω1,0 in a double complex. We find the natural metric, volume form, Hodge * operator, Laplace and Maxwell operators and projective module structure. We show that the q-monopole as spin connection induces a natural Levi-Civita type connection and find its Ricci curvature and q-Dirac operator . We find the possibility of an antisymmetric volume form quantum correction to the Ricci curvature and Lichnerowicz-type formulae for We also remark on the geometric q-Borel-Weil-Bott construction.
Similar content being viewed by others
References
Andersen, H.H., Polo, P., Wen. K.X.: Representations of quantum algebras. Invent. Math. 104, 1–59 (1991)
Bonechi, F., Ciccoli, N., Tarlini, M.: Noncommutative instantons in the 4-sphere from quantum groups. Commun. Math. Phys. 226, 419–432 (2002)
Bibikov, P.N., Kulish, P.P.: Dirac operators on the quantum group SU q (2) and the quantum sphere. J. Math. Sci. 100, 239–250 (2000)
Brzeziński, T., Majid, S.: Quantum group gauge theory on quantum spaces. Commun. Math. Phys. 157, 591–638 (1993) Erratum 167, 235 (1995)
Brzeziński, T., Majid, S.: Quantum differentials and the q-monopole revisited. Acta Appl. Math. 54, 185–232 (1998)
Connes, A.: Noncommutative Geometry. London: Academic Press, 1994
Dabrowski, L., Sitarz, A.: Dirac operator on the standard Podles sphere. In: Noncommutative geometry and quantum groups, P. Hajac, W. Pusz, (eds.), Banach Center Publications, 61, 49–58 (2003)
Hajac, P., Majid, S.: Projective module description of the q-monopole. Commun. Math. Phys. 206, 246–464 (1999)
Majid, S.: Foundations of Quantum Group Theory. Cambridge: Cambridge Univeristy Press, 1995
Majid, S.: Quantum and braided group Riemannian geometry. J. Geom. Phys. 30, 113–146 (1999)
Majid, S.: Riemannian Geometry of Quantum Groups and Finite Groups with Nonuniversal Differentials. Commun. Math. Phys. 225, 131–170 (2002)
Majid, S.: Ricci tensor and Dirac operator on ℂ q [SL2] at roots of unity. Lett. Math. Phys. 63, 39–54 (2003)
Owczarek, R.: Dirac operators on the Podles sphere. Int. J. Theor. Phys. 40, 163–170 (2001)
Parshall, B., Wang, S.: Quantum linear groups. Mem. Amer. Math. Soc. 89, (1991)
Pinzul, A., Stern, A.: Dirac operator on the quantum sphere. Phys. Lett. B 512, 217–224 (2001)
Podles, P.: Quantum spheres. Lett. Math. Phys. 14, 193–202 (1987)
Podles, P.: Differential calculus on quantum spheres. Lett. Math. Phys. 18, 107–119 (1989)
Podles, P.: The classification of differentical structures on quantum 2-spheres. Commun. Math. Phys. 150, 167–179 (1992)
Sweedler, M.: Hopf Algebras. New York: Benjamin Press (1969)
Schneider, H-J.: Principal homogeneous spaces for arbitrary Hopf algebras. Isr. J. Math. 72, 167–195 (1990)
Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups). Commun. Math. Phys. 122, 125–170 (1989)
Gover, A.R., Zhang, R.B.: Geometry of quantum homogeneous vector bundles and representation theory of quantum groups I (appendix of). Rev. Math. Phys. 11 533–552 (1999)
Author information
Authors and Affiliations
Additional information
Communicated by L. Takhtajan
Rights and permissions
About this article
Cite this article
Majid, S. Noncommutative Riemannian and Spin Geometry of the Standard q-Sphere. Commun. Math. Phys. 256, 255–285 (2005). https://doi.org/10.1007/s00220-005-1295-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-005-1295-8