Classical $\mathcal{W}$-algebras for centralizers

TL;DR

This paper introduces a new family of classical $ ext{W}$-algebras associated with the centralizer of nilpotent elements in $ ext{gl}_N$, providing explicit generators and establishing their isomorphism with the center of affine vertex algebras at the critical level.

Contribution

The authors define and analyze a novel class of Poisson vertex algebras $ ext{W}( ext{a})$ linked to centralizers of nilpotent elements, including explicit generators and structural properties.

Findings

$ ext{W}( ext{a})$ is a polynomial algebra in infinitely many variables.

Explicit free generators of $ ext{W}( ext{a})$ are constructed.

$ ext{W}( ext{a})$ is isomorphic to the center at the critical level of the affine vertex algebra.

Abstract

We introduce a new family of Poisson vertex algebras $\mathcal{W}(\mathfrak{a})$ analogous to the classical $\mathcal{W}$-algebras. The algebra $\mathcal{W}(\mathfrak{a})$ is associated with the centralizer $\mathfrak{a}$ of an arbitrary nilpotent element in $\mathfrak{gl}_N$. We show that $\mathcal{W}(\mathfrak{a})$ is an algebra of polynomials in infinitely many variables and produce its free generators in an explicit form. This implies that $\mathcal{W}(\mathfrak{a})$ is isomorphic to the center at the critical level of the affine vertex algebra associated with $\mathfrak{a}$.