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Non-weight modules over the Heisenberg–Virasoro algebra and the $W$ $W$ algebra $W (2, 2)$ $W (2, 2)$

Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, P. R. China

E-mail Address: hjchen@ustc.edu.cn

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and

Xiangqian Guo

Department of Mathematics, Zhengzhou University, Zhengzhou 450001, Henan, P. R. China

E-mail Address: guoxq@zzu.edu.cn

Corresponding author.

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https://doi.org/10.1142/S0219498817500979Cited by:34 (Source: Crossref)

Abstract

In this paper, we construct and study some non-weight modules for the Heisenberg–Virasoro algebra and the $W$ $W$ algebra $W (2, 2)$ $W (2, 2)$ . We determine the modules, whose restriction to the universal enveloping algebra of the degree- $0$ $0$ part (modulo center) are free of rank $1$ $1$ for these two algebras. In the most interesting case, this degree- $0$ $0$ part is not the Cartan subalgebra. We also determine the simplicity of these modules, which provide new simple modules for the $W$ $W$ algebra $W (2, 2)$ $W (2, 2)$ .

Communicated by M. Gorelik

Keywords:

AMSC: 17B10, 17B20, 17B65, 17B66, 17B68

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