Dong Jun Choi | Searcharxiv

arXiv2026

Poisson structures of deformed $W$-algebras and classical $W$-algebras

In this paper, we explain how classical $W$-algebras can be obtained as limits of $q$-deformed $W$-algebras in type $A$. To this end, we give a detailed examination of the construction of deformed $W$-algebras via Hamiltonian reduction of deformed affine Poisson algebras. We first derive an affine Poisson vertex algebra (PVA) from the Poisson structure of the deformed affine Poisson algebra. This construction identifies a specific set of generators of the deformed $W$-algebra with generators of the classical $W$-algebra.

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arXiv2026

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

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.

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