𝒲-Algebras Associated With Centralizers in Type A

Abstract

Abstract We introduce a new family of affine $\mathcal{W}$-algebras $\mathcal{W}^{k}(\mathfrak{a})$ associated with the centralizers of arbitrary nilpotent elements in $\mathfrak{gl}_N$. We define them by using a version of the BRST (Becchi, Rouet, Stora and Tyutin) complex of the quantum Drinfeld–Sokolov reduction. A family of free generators of $\mathcal{W}^{k}(\mathfrak{a})$ is produced in an explicit form. We also give an analogue of the Fateev–Lukyanov realization for the new $\mathcal{W}$-algebras by applying a Miura-type map.