Birational Equivalences and Generalized Weyl Algebras

TL;DR

This paper investigates the birational properties of generalized Weyl algebras (GWAs), computes Hochschild homologies of quantum groups and Podle's spheres, and addresses key problems in their classification and smoothness.

Contribution

It introduces methods to analyze GWAs via birational equivalences, solves the birational equivalence problem, and computes Hochschild homologies for quantum structures.

Findings

Hochschild homologies of quantum groups and Podle's spheres are computed after localization.

Every GWA is shown to be birationally equivalent to a smash product with a 1-torus.

The birational equivalence and smoothness problems for GWAs are solved.

Abstract

We calculate suitably localized Hochschild homologies of various quantum groups and Podle\'s spheres after realizing them as generalized Weyl algebras (GWAs). We use the fact that every GWA is birationally equivalent to a smash product with a 1-torus. We also address and solve the birational equivalence problem, and the birational smoothness problem for GWAs.