Noncommutative Differential Geometry of Generalized Weyl Algebras

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.