Classification of Harish-Chandra modules over the $W$-algebra W(2,2)
  In this paper, we classify all irreducible weight modules with finite dimensional weight spaces over the $W$-algebra $W(2, 2)$. Meanwhile, all indecomposable modules with one dimensional weight spaces over the $W$-algebra $W(2, 2)$ are also determined. 
Dong Liu and Linsheng Zhu