Restricted modules over some ℤ-graded Lie algebras

Abstract

In this paper, we answer a question asked by Kaiming Zhao at the mini course “Simple representations of the Virasoro algebra” at Xiamen mathematical center in the fall term of 2021. More precisely, we prove that if one special element in the Lie algebra $\mathfrak{g}$ (including the Virasoro algebra, the Heisenberg-Virasoro algebra and affine Virasoro algebras) acts locally nilpotently on an irreducible module V, then V is a restricted $\mathfrak{g}$-module.

Keywords:

• Locally nilpotent
• restricted module
• Virasoro algebra

2020 Mathematics Subject Classification:

• 17B10
• 17B20
• 17B65
• 17B66
• 17B68

Acknowledgments

The authors thank Prof. Zhao for many helpful discussions.

Additional information

Funding

D. Gao is partially supported by National Postdoctoral Program for Innovative Talents of China (grant 20220158), China Postdoctoral Science Foundation (grant 2022M711708), NSF of China (grant 11931009) and Nankai Zhide Foundation. C. Xu is partially supported by NSF of China (grants 11801375, 12261077).