Usually, I think $w_\infty$ is an Lie algebra, though it seems that $w_\infty$ can be viewed as an associative algebra. The algebra spanned by {$z^{\alpha_1}z^{\alpha_2}\partial_{z}^{\beta_1}\partial_{y}^{\beta_2}$} should be Lie algebra. "span" means as a linear basis.

thanks for the answer, which is very helpful. I want to know whether the following relation is right, $\delta(-\lambda,z+\mu-\lambda)=???\delta(-\mu,z)$ Thanks again.