On $W_{1+\infty}$ $n$-algebra
  We present the nontrivial $W_{1+\infty}$ $n$-algebra and analyze its remarkable properties. We investigate the $W_{1+\infty}$ $n$-algebra in the Landau problem and discuss the realization of the classical $w_{\infty}$ 3-algebra. Furthermore, we discuss the case of the many-body system in the lowest Landau level and derive the constraints for correlation functions of the vertex operators. 
Chun-Hong Zhang, Lu Ding, Zhao-Wen Yan, Ke Wu, Wei-Zhong Zhao
hep-th