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arXiv · 2206.03464

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

Xiangui Zhao
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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.

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Xiangui Zhao. 2022-06-07. Growth of generalized Weyl algebras over polynomial algebras and Laurent polynomial algebras. https://doi.org/10.1007/s11425-022-1992-2

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