On the category $\mathcal{O}$ for generalized Weyl algebras

TL;DR

This paper proves that for a generalized Weyl algebra $H(R,\phi,z)$ with a central nilpotent perturbation and an $l$-step twist, the category $\mathcal{O}(H(R,\phi,z))$ is equivalent to $\mathcal{O}(H(R,\phi^l,z))$ via the functor $M\mapsto M^{z^{\infty}}$. The authors develop a Morita-theoretic framework using completions and idempotent lifting to identify a corner algebra with a twisted GWA and then employ a Frobenius-type map $Fr_l$ to construct an exact equivalence, with an explicit description of the $R$-module structure under the twist. This generalizes known results for the Weyl algebra over finite rings and yields concrete applications to noncommutative deformations of type A Kleinian singularities and quantized Weyl algebras, including characteristic-$p$ analogues and a simple-dimension result for objects in $\mathcal{O}$. Overall, the work clarifies how $\mathcal{O}$-modules behave under $l$-fold twisting and provides a robust method to transfer representation-theoretic information across twists.

Abstract

Let $H(R, φ, z)$ be a generalized Weyl algebra associated with a ring $R$, its central element $z\in Z(R)$ and an automorphism $φ,$ such that for some $l \geq 1$, $φ^l(z)-z$ is nilpotent and $(z,φ^i(z))=R$ for all $0<i<l$. We prove that the category $\mathcal{O}$ over $H(R, z,φ)$ is equivalent to the category $\mathcal{O}$ over its $l$-th twist the generalized Weyl algebra $H(R, z,φ^l).$ This result is significantly more general than the corresponding one for the Weyl algebra over $\mathbb{Z}/p^n\mathbb{Z}.$

Theorems & Definitions (22)

• Theorem 1.1
• Definition 1.1
• Theorem 1.2
• Proposition 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof