Weak generators of W-algebras for type $A$
Min Hee Park, Uhi Rinn Suh
TL;DR
The paper studies weak generating sets for type A W-algebras, both classical $\mathcal{W}^k(\mathfrak{g},f)$ and quantum $W^k(\mathfrak{g},f)$, with $\mathfrak{g}=\mathfrak{sl}_N$ or $\mathfrak{sl}_{N_1|N_2}$ and even nilpotent $f$ described by partitions $(m_1,\dots,m_d)$. It constructs two main families of weak generators for the classical algebras: a big-weight set $\mathcal{C}^f_{\text{big}}$ and a small-weight set $\mathcal{C}^f_{\text{sm}}$, whose elements are derived from a basis of the centralizer $\mathfrak{g}^f$ and mapped by the omega isomorphism. The quantum case is handled via the $\varepsilon$-deformed BRST framework, showing that the corresponding $W^{\varepsilon^{-1}k}(\mathfrak{g},f)$ is weakly generated for generic $\varepsilon$ by the quantum analogues of the classical weak generators. The results cover a broad range of $f$ (including rectangular and minimal nilpotents) and provide explicit constructions in several examples, offering practical routes to obtain all strong generators from a compact weak generating set. This advances the understanding of the generation structure of W-algebras and highlights the classical-quantum correspondence through weak generation patterns.
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.