Generalized finite and affine $W$-algebras in type $A$

TL;DR

This work develops a unified framework of generalized finite and affine W-algebras $W^k(\lambda,\mu)$ attached to centralizers of nilpotent elements in $\mathfrak{gl}_N$, parameterized by partitions $\lambda$ of $N$ and $\mu$ of $n$. The construction uses a BRST complex $C^k(\lambda,\mu)$ for a quantum Drinfeld--Sokolov reduction, yielding affine algebras that interpolate between known families and recover classical W-algebras in special cases; the Zhu functor then links these to generalized finite W-algebras $U(\lambda,\mu)$, with explicit structural descriptions. The main results show that $Zhu_H(W^k(\lambda,\mu))\cong U(\lambda,\mu)$, with two realizations via Lie algebra cohomology and graded centers, and provide concrete generators and dimensions in principal and minimal nilpotent cases. In the principal nilpotent setting, $U(\lambda,(n))$ is the center of $U(\mathfrak{a})$ (hence commutative), while the minimal nilpotent case yields explicit weight-1 and weight-2 generators for both finite and affine algebras, enabling detailed structural understanding. Overall, the paper offers a coherent, interlocking picture of generalized W-algebras across a spectrum of centralizers, including explicit generators and cohomological descriptions that connect affine and finite theories.

Abstract

We construct a new family of affine $W$-algebras $W^k(λ,μ)$ parameterized by partitions $λ$ and $μ$ associated with the centralizers of nilpotent elements in $\mathfrak{gl}_N$. The new family unifies a few known classes of $W$-algebras. In particular, for the column-partition $λ$ we recover the affine $W$-algebras $W^k(\mathfrak{gl}_N,f)$ of Kac, Roan and Wakimoto, associated with nilpotent elements $f\in\mathfrak{gl}_N$ of type $μ$. Our construction is based on a version of the BRST complex of the quantum Drinfeld-Sokolov reduction. We show that the application of the Zhu functor to the vertex algebras $W^k(λ,μ)$ yields a family of generalized finite $W$-algebras $U(λ,μ)$ which we also describe independently as associative algebras.

Theorems & Definitions (47)

• Definition 2.1
• Example 2.2
• Example 2.3
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Definition 3.3
• Lemma 3.4
• proof