
SIGMA 12 (2016), 059, 18 pages arXiv:1602.07456
https://doi.org/10.3842/SIGMA.2016.059
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, 15245 Białystok, Poland
Received February 29, 2016, in final form June 14, 2016; Published online June 23, 2016
Abstract
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 zerodegree 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 threedimensional firstorder 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 firstorder 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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References

Bavula V., Tensor homological minimal algebras, global dimension of the tensor product of algebras and of generalized Weyl algebras, Bull. Sci. Math. 120 (1996), 293335.

Beggs E.J., Majid S., Spectral triples from bimodule connections and Chern connections, arXiv:1508.04808.

Beggs E.J., Smith S.P., Noncommutative complex differential geometry, J. Geom. Phys. 72 (2013), 733, arXiv:1209.3595.

Brzeziński T., Noncommutative connections of the second kind, J. Algebra Appl. 7 (2008), 557573, arXiv:0802.0445.

Brzeziński T., Circle and line bundles over generalized Weyl algebras, Algebr. Represent. Theory 19 (2016), 5769, arXiv:1405.3105.

Brzeziński T., El Kaoutit L., Lomp C., Noncommutative integral forms and twisted multiderivations, J. Noncommut. Geom. 4 (2010), 289312, arXiv:0901.2710.

Brzeziński T., Fairfax S.A., Quantum teardrops, Comm. Math. Phys. 316 (2012), 151170, arXiv:1107.1417.

Brzeziński T., Hajac P.M., The ChernGalois character, C. R. Math. Acad. Sci. Paris 338 (2004), 113116, math.KT/0306436.

Brzeziński T., Majid S., Quantum group gauge theory on quantum spaces, Comm. Math. Phys. 157 (1993), 591638, Erratum, Comm. Math. Phys. 167 (1995), 235235, hepth/9208007.

Brzeziński T., Sitarz A., Smooth geometry of the noncommutative pillow, cones and lens spaces, J. Noncommut. Geom., to appear, arXiv:1410.6587.

Connes A., Noncommutative geometry and reality, J. Math. Phys. 36 (1995), 61946231.

Dade E.C., Compounding Clifford's theory, Ann. of Math. 91 (1970), 236290.

Dade E.C., Groupgraded rings and modules, Math. Z. 174 (1980), 241262.

Hajac P.M., Strong connections on quantum principal bundles, Comm. Math. Phys. 182 (1996), 579617, hepth/9406129.

Hong J.H., Szymański W., Quantum lens spaces and graph algebras, Pacific J. Math. 211 (2003), 249263.

Khalkhali M., Landi G., van Suijlekom W.D., Holomorphic structures on the quantum projective line, Int. Math. Res. Not. 2011 (2011), 851884, arXiv:0907.0154.

Krähmer U., On the Hochschild (co)homology of quantum homogeneous spaces, Israel J. Math. 189 (2012), 237266.

Liu L., Homological smoothness and deformations of generalized Weyl algebras, Israel J. Math. 209 (2015), 949992, arXiv:1304.7117.

Lunts V.A., Rosenberg A.L., Kashiwara theorem for hyperbolic algebras, Preprint MPIM199982, 1999.

Majid S., Noncommutative Riemannian and spin geometry of the standard $q$sphere, Comm. Math. Phys. 256 (2005), 255285, math.QA/0307351.

Năstăsescu C., van Oystaeyen F., Graded ring theory, NorthHolland Mathematical Library, Vol. 28, NorthHolland Publishing Co., Amsterdam  New York, 1982.

Podleś P., Quantum spheres, Lett. Math. Phys. 14 (1987), 193202.

van den Bergh M., A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc. 126 (1998), 13451348.

Woronowicz S.L., Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613665.

Woronowicz S.L., Twisted ${\rm SU}(2)$ group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), 117181.

