Modules over the Heisenberg-Virasoro and $W(2,2)$ algebras
  In this paper, we consider the modules for the Heisenberg-Virasoro algebra and the W algebra $W(2,2)$. We determine the modules whose restriction to the Cartan subalgebra (modulo center) are free of rank $1$ for the two algebras. We also determine the simplicity of these modules. These modules provide new simple modules for the W algebra $W(2,2)$. 
Hongjia Chen and Xiangqian Guo
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