Weak generators of W-algebras for type $A$

In this paper, we find weak generating sets for a classical W-algebra ${\mathcal{W}}^k(\mathfrak{g},f)$ when $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_N$ or ${\mathfrak{s}\mathfrak{l}}_{N_1\mathrm{\mid }{N}_2}$. Furthermore, observing the relation between quantum and classical W-algebras, we further derive crucial information about the weak generating sets of quantum W-algebras at generic levels.