Generalized Finite and Affine W -Algebras in Type A

Abstract We construct a new family of affine $W$-algebras $W^{k}(\lambda ,\mu )$ parameterized by partitions $\lambda $ and $\mu $ associated with the centralizers of nilpotent elements in $\mathfrak{g}\mathfrak{l}_{N}$. The new family unifies a few known classes of $W$-algebras. In particular, for the column-partition $\lambda $ we recover the affine $W$-algebras $W^{k}(\mathfrak{g}\mathfrak{l}_{N},f)$ of Kac, Roan, and Wakimoto, associated with nilpotent elements $f\in \mathfrak{g}\mathfrak{l}_{N}$ of type $\mu $. 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 $W^{k}(\lambda ,\mu )$ yields a family of generalized finite $W$-algebras $U(\lambda ,\mu )$, which we also describe independently as associative algebras.