Circle and line bundles over generalized Weyl algebras

Strongly $\mathbb{Z}$-graded algebras or principal circle bundles and associated line bundles or invertible bimodules over a class of generalized Weyl algebras $\mathcal{B}(p;q,0)$ (over a ring of polynomials in one variable) are constructed. The Chern-Connes pairing between the cyclic cohomology of $\mathcal{B}(p;q,0)$ and the isomorphism classes of sections of associated line bundles over $\mathcal{B}(p;q,0)$ is computed thus demonstrating that these bundles, which are labeled by integers, are non-trivial and mutually non-isomorphic. The constructed strongly $\mathbb{Z}$-graded algebras are shown to have Hochschild cohomology reminiscent of that of Calabi-Yau algebras. The paper is supplemented by an observation that a grading by an Abelian group in the middle of a short exact sequence is strong if and only if the induced gradings by the outer groups in the sequence are strong.