Weak generators of W-algebras for type $A$

TL;DR

This paper identifies weak generating sets for classical W-algebras associated with type A Lie algebras and explores their quantum counterparts, providing new insights into their structure and relations.

Contribution

It introduces explicit weak generating sets for classical W-algebras of type A and analyzes their quantum versions at generic levels, advancing understanding of their algebraic structure.

Findings

Explicit weak generators for classical W-algebras of type A

Relations between quantum and classical W-algebras clarified

Insights into the structure of quantum W-algebras at generic levels

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.