Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 10 (2014), 015, 8 pages      arXiv:1311.4758      https://doi.org/10.3842/SIGMA.2014.015
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

On the Smoothness of the Noncommutative Pillow and Quantum Teardrops

Tomasz Brzeziński
Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, UK

Received December 03, 2013, in final form February 09, 2014; Published online February 14, 2014

Abstract
Recent results by Krähmer [Israel J. Math. 189 (2012), 237-266] on smoothness of Hopf-Galois extensions and by Liu [arXiv:1304.7117] on smoothness of generalized Weyl algebras are used to prove that the coordinate algebras of the noncommutative pillow orbifold [Internat. J. Math. 2 (1991), 139-166], quantum teardrops ${\mathcal O}({\mathbb W}{\mathbb P}_q(1,l))$ [Comm. Math. Phys. 316 (2012), 151-170], quantum lens spaces ${\mathcal O}(L_q(l;1,l))$ [Pacific J. Math. 211 (2003), 249-263], the quantum Seifert manifold ${\mathcal O}(\Sigma_q^3)$ [J. Geom. Phys. 62 (2012), 1097-1107], quantum real weighted projective planes ${\mathcal O}({\mathbb R}{\mathbb P}_q^2(l;\pm))$ [PoS Proc. Sci. (2012), PoS(CORFU2011), 055, 10 pages] and quantum Seifert lens spaces ${\mathcal O}(\Sigma_q^3(l;-))$ [Axioms 1 (2012), 201-225] are homologically smooth in the sense that as their own bimodules they admit finitely generated projective resolutions of finite length.

Key words: smooth algebra; generalized Weyl algebra; strongly graded algebra; noncommutative pillow; quantum teardrop; quantum lens space; quantum real weighted projective plane.

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References

  1. Bavula V., Tensor homological minimal algebras, global dimension of the tensor product of algebras and of generalized Weyl algebras, Bull. Sci. Math. 120 (1996), 293-335.
  2. Bratteli O., Elliott G.A., Evans D.E., Kishimoto A., Noncommutative spheres. I, Internat. J. Math. 2 (1991), 139-166.
  3. Brzeziński T., Circle actions on a quantum Seifert manifold, PoS Proc. Sci. (2012), PoS(CORFU2011), 055, 10 pages, arXiv:1203.6801.
  4. Brzeziński T., Fairfax S.A., Bundles over quantum real weighted projective spaces, Axioms 1 (2012), 201-225, arXiv:1207.2313.
  5. Brzeziński T., Fairfax S.A., Quantum teardrops, Comm. Math. Phys. 316 (2012), 151-170, arXiv:1107.1417.
  6. Brzeziński T., Hajac P.M., The Chern-Galois character, C. R. Math. Acad. Sci. Paris 338 (2004), 113-116, math.KT/0306436.
  7. Brzeziński T., Zieliński B., Quantum principal bundles over quantum real projective spaces, J. Geom. Phys. 62 (2012), 1097-1107, arXiv:1105.5897.
  8. Dąbrowski L., Grosse H., Hajac P.M., Strong connections and Chern-Connes pairing in the Hopf-Galois theory, Comm. Math. Phys. 220 (2001), 301-331, math.QA/9912239.
  9. Evans D.E., Kawahigashi Y., Quantum symmetries on operator algebras, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1998.
  10. Goodearl K.R., Zhang J.J., Homological properties of quantized coordinate rings of semisimple groups, Proc. Lond. Math. Soc. (3) 94 (2007), 647-671, math.QA/0510420.
  11. Hajac P.M., Strong connections on quantum principal bundles, Comm. Math. Phys. 182 (1996), 579-617, hep-th/9406129.
  12. Hong J.H., Szymański W., Quantum lens spaces and graph algebras, Pacific J. Math. 211 (2003), 249-263.
  13. Krähmer U., On the Hochschild (co)homology of quantum homogeneous spaces, Israel J. Math. 189 (2012), 237-266, arXiv:0806.0267.
  14. Liu L.-Y., Homological smoothness and deformations of generalized Weyl algebras, arXiv:1304.7117.
  15. McConnell J.C., Robson J.C., Noncommutative Noetherian rings, Graduate Studies in Mathematics, Vol. 30, American Mathematical Society, Providence, RI, 2001.
  16. Năstăsescu C., van Oystaeyen F., Graded ring theory, North-Holland Mathematical Library, Vol. 28, North-Holland Publishing Co., Amsterdam, 1982.
  17. Podleś P., Quantum spheres, Lett. Math. Phys. 14 (1987), 193-202.
  18. Rieffel M.A., $C^{\ast}$-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415-429.
  19. Thurston W.P., The geometry and topology of three-manifolds, Princeton University, 1980, available at http://www.msri.org/publications/books/gt3m/.
  20. van den Bergh M., A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc. 126 (1998), 1345-1348, Erratum, Proc. Amer. Math. Soc. 130 (2002), 2809-2810.
  21. Woronowicz S.L., Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613-665.


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