Skew derivations on generalized Weyl algebras

TL;DR

This paper systematically constructs and classifies skew derivations on generalized Weyl algebras, analyzing their properties, compatibility with grading, and conditions for orthogonality, with explicit examples over polynomial rings.

Contribution

It provides a comprehensive construction and classification of skew derivations on generalized Weyl algebras, including conditions for $Q$-skew derivations and orthogonality.

Findings

Constructed wide classes of skew derivations on degree-one generalized Weyl algebras.

Determined conditions for skew derivations to be $Q$-skew.

Classified skew derivations over polynomial rings with linear central elements.

Abstract

A wide class of skew derivations on degree-one generalized Weyl algebras $R(a,\varphi)$ over a ring $R$ is constructed. All these derivations are twisted by a degree-counting extensions of automorphisms of $R$. It is determined which of the constructed derivations are $Q$-skew derivations. The compatibility of these skew derivations with the natural ${\mathbb Z}$-grading of $R(a,\varphi)$ is studied. Additional classes of skew derivations are constructed for generalized Weyl algebras given by an automorphism $\varphi$ of a finite order. Conditions that the central element $a$ that forms part of the structure of $R(a,\varphi)$ need to satisfy for the orthogonality of pairs of aforementioned skew derivations are derived. General constructions are illustrated by classification of skew derivations of generalized Weyl algebras over the polynomial ring in one variable and with a linear polynomial as the central element.