Growth of generalized Weyl algebras over polynomial algebras and Laurent polynomial algebras

TL;DR

This paper investigates the growth and Gelfand-Kirillov dimension of generalized Weyl algebras over polynomial and Laurent polynomial algebras, providing conditions for their dimensions and a dichotomy in specific cases.

Contribution

It establishes necessary and sufficient conditions for the GK-dimension of GWAs, including a dichotomy for GWAs over polynomial algebras in two variables, extending previous results.

Findings

GK-dimension of GWAs over polynomial algebras is either 3 or infinite.

Provided criteria for when GK-dimension equals GK-dimension of base algebra plus one.

Results applicable to analyzing growth, simplicity, and cancellation properties of GWAs.

Abstract

We mainly study the growth and Gelfand-Kirillov dimension (GK-dimension) of generalized Weyl algebra (GWA) $A=D(\sigma,a)$ where $D$ is a polynomial algebra or a Laurent polynomial algebra. Several necessary and sufficient conditions for $\operatorname{GKdim}(A)=\operatorname{GKdim}(D)+1$ are given. In particular, we prove a dichotomy of the GK-dimension of GWAs over the polynomial algebra in two indeterminates, namely, $\operatorname{GKdim}(A)$ is either $3$ or $\infty$ in this case. Our results generalize several ones in the literature and can be applied to determine the growth, GK-dimension, simplicity, and cancellation properties of some GWAs.