Derivations of Quantum and Involution Generalized Weyl Algebras

TL;DR

This paper classifies derivations of degree-one generalized Weyl algebras over Laurent polynomial rings, including special cases like the Weyl-Hayashi algebra and certain localizations, advancing understanding of their algebraic structure.

Contribution

It provides a comprehensive classification of derivations for a broad class of generalized Weyl algebras, including notable special cases.

Findings

Classified derivations of degree-one generalized Weyl algebras

Covered the Weyl-Hayashi algebra as a special case

Analyzed algebras that localize to the group algebra of an infinite group

Abstract

We classify the derivations of degree-one generalized Weyl algebras over a univariate Laurent polynomial ring. In particular, our results cover the Weyl-Hayashi algebra, a quantization of the first Weyl algebra arising as a primitive factor algebra of $U_q^+(\mathfrak{so}_5)$, and a family of algebras which localize to the group algebra of the infinite group with generators $x$ and $y$, subject to the relation $xy = y^{-1}x.$