Title: On the category $\mathcal{O}$ for generalized Weyl algebras Show Chinese title

Title: 关于广义Weyl代数的范畴$\mathcal{O}$

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

Abstract: 设 $H(R, \phi, z)$ 是与环 $R$、其中心元素 $z\in Z(R)$ 和自同构 $\phi,$ 相关的广义Weyl代数，使得对于某些 $l \geq 1$， $\phi^l(z)-z$ 是幂零的且 $(z,\phi^i(z))=R$ 对所有 $0<i<l$ 成立。 我们证明了在 $H(R, z,\phi)$ 上的范畴 $\mathcal{O}$ 与它的 $l$-次扭变的广义 Weyl 代数 $H(R, z,\phi^l).$ 上的范畴 $\mathcal{O}$ 是等价的。这个结果比在 $\mathbb{Z}/p^n\mathbb{Z}.$ 上的 Weyl 代数的相应结果要普遍得多。