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

SIGMA 12 (2016), 059, 18 pages      arXiv:1602.07456

Noncommutative Differential Geometry of Generalized Weyl Algebras

Tomasz Brzeziński ab
a) Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, UK
b) Department of Mathematics, University of Białystok, K. Ciołkowskiego 1M, 15-245 Białystok, Poland

Received February 29, 2016, in final form June 14, 2016; Published online June 23, 2016

Elements of noncommutative differential geometry of ${\mathbb Z}$-graded generalized Weyl algebras ${\mathcal A}(p;q)$ over the ring of polynomials in two variables and their zero-degree subalgebras ${\mathcal B}(p;q)$, which themselves are generalized Weyl algebras over the ring of polynomials in one variable, are discussed. In particular, three classes of skew derivations of ${\mathcal A}(p;q)$ are constructed, and three-dimensional first-order differential calculi induced by these derivations are described. The associated integrals are computed and it is shown that the dimension of the integral space coincides with the order of the defining polynomial $p(z)$. It is proven that the restriction of these first-order differential calculi to the calculi on ${\mathcal B}(p;q)$ is isomorphic to the direct sum of degree 2 and degree $-2$ components of ${\mathcal A}(p;q)$. A Dirac operator for ${\mathcal B}(p;q)$ is constructed from a (strong) connection with respect to this differential calculus on the (free) spinor bimodule defined as the direct sum of degree 1 and degree $-1$ components of ${\mathcal A}(p;q)$. The real structure of ${\rm KO}$-dimension two for this Dirac operator is also described.

Key words: generalized Weyl algebra; skew derivation; differential calculus; principal comodule algebra; strongly graded algebra; Dirac operator.

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  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. Beggs E.J., Majid S., Spectral triples from bimodule connections and Chern connections, arXiv:1508.04808.
  3. Beggs E.J., Smith S.P., Non-commutative complex differential geometry, J. Geom. Phys. 72 (2013), 7-33, arXiv:1209.3595.
  4. Brzeziński T., Non-commutative connections of the second kind, J. Algebra Appl. 7 (2008), 557-573, arXiv:0802.0445.
  5. Brzeziński T., Circle and line bundles over generalized Weyl algebras, Algebr. Represent. Theory 19 (2016), 57-69, arXiv:1405.3105.
  6. Brzeziński T., El Kaoutit L., Lomp C., Non-commutative integral forms and twisted multi-derivations, J. Noncommut. Geom. 4 (2010), 289-312, arXiv:0901.2710.
  7. Brzeziński T., Fairfax S.A., Quantum teardrops, Comm. Math. Phys. 316 (2012), 151-170, arXiv:1107.1417.
  8. Brzeziński T., Hajac P.M., The Chern-Galois character, C. R. Math. Acad. Sci. Paris 338 (2004), 113-116, math.KT/0306436.
  9. Brzeziński T., Majid S., Quantum group gauge theory on quantum spaces, Comm. Math. Phys. 157 (1993), 591-638, Erratum, Comm. Math. Phys. 167 (1995), 235-235, hep-th/9208007.
  10. Brzeziński T., Sitarz A., Smooth geometry of the noncommutative pillow, cones and lens spaces, J. Noncommut. Geom., to appear, arXiv:1410.6587.
  11. Connes A., Noncommutative geometry and reality, J. Math. Phys. 36 (1995), 6194-6231.
  12. Dade E.C., Compounding Clifford's theory, Ann. of Math. 91 (1970), 236-290.
  13. Dade E.C., Group-graded rings and modules, Math. Z. 174 (1980), 241-262.
  14. Hajac P.M., Strong connections on quantum principal bundles, Comm. Math. Phys. 182 (1996), 579-617, hep-th/9406129.
  15. Hong J.H., Szymański W., Quantum lens spaces and graph algebras, Pacific J. Math. 211 (2003), 249-263.
  16. Khalkhali M., Landi G., van Suijlekom W.D., Holomorphic structures on the quantum projective line, Int. Math. Res. Not. 2011 (2011), 851-884, arXiv:0907.0154.
  17. Krähmer U., On the Hochschild (co)homology of quantum homogeneous spaces, Israel J. Math. 189 (2012), 237-266.
  18. Liu L., Homological smoothness and deformations of generalized Weyl algebras, Israel J. Math. 209 (2015), 949-992, arXiv:1304.7117.
  19. Lunts V.A., Rosenberg A.L., Kashiwara theorem for hyperbolic algebras, Preprint MPIM-1999-82, 1999.
  20. Majid S., Noncommutative Riemannian and spin geometry of the standard $q$-sphere, Comm. Math. Phys. 256 (2005), 255-285, math.QA/0307351.
  21. Năstăsescu C., van Oystaeyen F., Graded ring theory, North-Holland Mathematical Library, Vol. 28, North-Holland Publishing Co., Amsterdam - New York, 1982.
  22. Podleś P., Quantum spheres, Lett. Math. Phys. 14 (1987), 193-202.
  23. van den Bergh M., A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc. 126 (1998), 1345-1348.
  24. Woronowicz S.L., Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613-665.
  25. Woronowicz S.L., Twisted ${\rm SU}(2)$ group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), 117-181.

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