Circle and Line Bundles Over Generalized Weyl Algebras

Strongly -graded algebras or principal circle bundles and associated line bundles or invertible bimodules over a class of generalized Weyl algebras (over a ring of polynomials in one variable) are constructed. The Chern-Connes pairing between the cyclic cohomology of and the isomorphism classes of sections of associated line bundles over is computed, thus demonstrating that these bundles, which are labelled by integers, are non-trivial and mutually non-isomorphic. The constructed strongly -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.