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

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 $\mathrm{GKdim}(A)=\mathrm{GKdim}(D)+1$ are given. In particular, we prove a dichotomy of the GK-dimension of GWAs over the polynomial algebra in two indeterminates, namely, $\mathrm{GKdim}(A)$ 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.

22 pages. Accepted for publication in SCIENCE CHINA Mathematics