Weak generators of W-algebras for type $A$

Abstract

In this paper, we find weak generating sets for a classical W-algebra ${\mathcal{W}}^k(\mathfrak{g},f)$ when $\mathfrak{g}={\mathfrak{sl}}_N$ or ${\mathfrak{sl}}_{N_1|N_2}$. 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.