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

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, 15-245 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 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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