Circle and line bundles over generalized Weyl algebras

TL;DR

This paper constructs and analyzes circle and line bundles over generalized Weyl algebras, demonstrating their non-triviality and computing their topological invariants, with implications for Hochschild cohomology and algebraic gradings.

Contribution

It introduces strongly graded algebras over generalized Weyl algebras and computes their topological invariants, revealing their non-trivial bundle structures.

Findings

Bundles are labeled by integers and are non-trivial.

Hochschild cohomology resembles that of Calabi-Yau algebras.

A grading by an Abelian group is strong iff induced gradings are strong.

Abstract

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.