Lifting free modules to generalized Weyl algebras

TL;DR

The paper develops a comprehensive framework for modules that are free over the base ring R for generalized Weyl algebras A=R(σ,a). It classifies all finite-rank R-free A-modules, starting with rank-one modules V_p parametrized by divisors p of a, and then extends to higher rank via a Smith normal form stratification. It provides sharp simplicity criteria, submodule structures, and composition series in the PID case, and connects these to weight modules with finite support. The results yield new simple modules for diverse algebras (including sl_2, Kleinian singularities deformations, the Weyl algebra A_1, and GL_q(2)) and produce explicit higher-rank Cartan-free modules, broadening the landscape of Cartan-free representations across GWAs and related algebras.

Abstract

We study modules over a generalized Weyl algebra $R(σ,a)$ which are free when restricted to the base ring $R$. When $R$ is an integral domain, we construct all such finite-rank modules up to isomorphism, leading to new simple modules over a variety of algebras. In particular, we show that free modules that have rank $1$ over $R$ can be parametrized as $V_{\mathsf{p}}$ where $\mathsf{p}$ is a divisor of $a$. We give simplicity criteria for $V_{\mathsf{p}}$ and, additionally, when $R$ is a PID, provide a complete combinatorial description of the submodule structure of $V_{\mathsf{p}}$ and of the weight modules occurring as subquotients. We also show that, under some mild conditions on $R(σ,a)$, there exist simple $R$-free modules of arbitrary finite rank. We apply our results to $\mathfrak{sl}_2$ in order to construct new families of simple Cartan-free modules of all finite ranks.

Theorems & Definitions (74)

• Definition 1
• Remark 1.1
• Example 1.2: The enveloping algebra of $\mathfrak{sl}_2$
• Example 1.3: The minimal primitive quotients of $U(\mathfrak{sl}_2$)
• Example 1.4: The Smith algebras
• Lemma 1.5
• proof
• Corollary 1.6
• Lemma 1.7
• proof