Birational Equivalences and Generalized Weyl Algebras

Atabey Kaygun

Istanbul Technical University, Istanbul, Turkey

Abstract: We calculate suitably localized Hochschild homologies of various quantum groups and Podleś 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.

Keywords: noncommutative birational equivalences, generalized Weyl algebras, Hochschild homology, Podleś spheres, quantum groups, quantum enveloping algebras.

MSC: 17B37, 16E40, 14E99

Received: June 24, 2024; in final form July 21, 2025; Published online July 30, 2025

Language: English

DOI: 10.3842/SIGMA.2025.063

ArXiv: 2009.14801