  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.
