Abstract
We describe noncommutative geometric aspects of twisted deformations, in particular of the spheres of Connes and Landi and of Connes and Dubois Violette, by using the differential and integral calculus on these spaces that is covariant under the action of their corresponding quantum symmetry groups. We start from multiparametric deformations of the orthogonal groups and related planes and spheres. We show that only in the twisted limit of these multiparametric deformations the covariant calculus on the plane gives, by a quotient procedure, a meaningful calculus on the sphere. In this calculus, the external algebra has the same dimension as the classical one. We develop the Haar functional on spheres and use it to define an integral of forms. In the twisted limit (differently from the general multiparametric case), the Haar functional is a trace and we thus obtain a cycle on the algebra. Moreover, we explicitly construct the *-Hodge operator on the space of forms on the plane and then by quotient on the sphere. We apply our results to even spheres and compute the Chern–Connes pairing between the character of this cycle, i.e. a cyclic 2n-cocycle, and the instanton projector defined in math.QA/0107070.
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References
Aschieri, P. and Castellani, L.: An Introduction to noncommutative differential geometry on quantum groups, Internat. J. Modern Phys. A 8 (1993), 1667 [arXiv:hep-th/9207084].
Aschieri, P. and Castellani, L.: Bicovariant calculus on twisted ISO(N), quantum Poincaré group and quantum Minkowski space, Internat. J. Modern Phys. A 11 (1996), 4513 [q-alg/9601006].
Aschieri, P., Castellani, L. and Scarfone, A. M.: Quantum orthogonal planes: ISO q,r(n + 1, n- 1) and SO q,r(n + 1,n- 1) bicovariant calculi, Eur. Phys. J. C 7 (1999), 159 [q-alg/9709032].
Bonechi, F., Ciccoli, N. and Tarlini, M.: Noncommutative instantons on the 4-sphere from quantum groups, math.qa/0012236.
Carow-Watamura, U., Schlieker, M. and Watamura, S.: SO-q(N) covariant differential calculus on quantum space and quantum deformation of Schrödinger equation, Z. Phys. C 49 (1991), 439.
Connes, A.: Noncommutative Geometry, Academic Press, New York, 1994.
Connes, A. and Dubois-Violette, M.:Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples, math.QA/0107070.
Connes, A. and Landi, G.: Noncommutative manifolds: The instanton algebra and isospectral deformations, Comm. Math. Phys. 221 (2001), 141 [math.qa/0011194].
Dobrev, V. K.: Canonical q deformations of noncompact Lie (super)algebras, J. Phys. A 26 (1993), 1317.
Faddeev, L. D., Reshetikhin, N. Y. and Takhtajan, L. A.: Quantization of Lie groups and Lie algebras, Lengingrad Math. J. 1 (1990), 193 [Algebra Anal. 1 (1990), 178].
Fiore, G.: Quantum groups SOq(N), Spq(n) have q-determinants, too, J. Phys. A 27 (1994), 3795.
Gracia-Bondia, J. M., Varilly, J. C. and Figueroa, H.: Elements of Noncommutative Geometry, Birkhaüser, Boston, 2001.
Jurco, B.: Differential calculus on quantized simple Lie groups, Lett. Math. Phys. 22 (1991), 177; Carow-Watamura, U., Schlieker, M., Watamura, S. and Weich, W.: Bicovariant and differential calculus on quantum groups SUq(n) and SOq(n), Comm. Math. Phys. 142 (1991), 605; Jurco, B.: Differential calculus on quantum groups: Constructive procedure, In: L. Castellani, and J. Wess (eds), Proc. International School of Physics 'Enrico Fermi', Varenna 1994, Course CXXVII: Quantum Groups and their Application in Physics, IOS/Ohmsha Press, 1996 [hep-th/9408179].
Loday, J. L.: Cyclic Homology. Springer-Verlag, Berlin, 1992.
Masuda, T., Nagakami, Y. and Watanabe, J.: Noncommutative differential geometry on the quantum two sphere of Podle\(s`\). I; an algebraic viewpoint, K-Theory 5 (1991), 151.
Ogievetsky, O. and Zumino, B.: Reality in the differential calculus on q Euclidean spaces, Lett. Math. Phys. 25 (1992), 121 [hep-th/9205003].
Podle\(s`\), P.: Quantum Spheres, Lett. Math. Phys. 14 (1987), 193.
Pusz, W. and Woronowicz, S. L.: Twisted second quantization, Rep. Math. Phys. 27 (1990), 231; Wess, J. and Zumino, B.: Covariant differential calculus on the quantum hyperplane, Nuclear Phys. Proc. Suppl. 18B (1991), 302.
Reshetikhin, N.: Multiparametric quantum groups and twisted quasitriangular Hopf algebras, Lett. Math. Phys. 20 (1990), 331; Schirrmacher, A.: Multiparameter R matrices and its quantum groups, J. Phys. A 24 (1991), L1249.
Sitarz, A.: Twists and spectral triples for isospectral deformations, math.QA/0102074
Steinacker, H.: Integration on quantum Euclidean space and sphere in N-dimensions, J. Math. Phys. 37 (1996), 7438 [q-alg/9506020].
Twietmeyer, E.: Real forms of Uq(g), Lett. Math. Phys. 24 (1992), 49; Aschieri, P.: Real forms of quantum orthogonal groups, q-Lorentz groups in any dimension, Lett. Math. Phys. 49 (1999), 1 [math.qa/9805120].
Varilly, J. C.: Quantum symmetry groups of noncommutative spheres, Comm. Math. Phys. 221 (2001), 511 [math.qa/0102065].
Woronowicz, S. L.: Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122 (1989), 125.
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Aschieri, P., Bonechi, F. On the Noncommutative Geometry of Twisted Spheres. Letters in Mathematical Physics 59, 133–156 (2002). https://doi.org/10.1023/A:1014942018467
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DOI: https://doi.org/10.1023/A:1014942018467