Generalized finite and affine $W$-algebras in type $A$
Generalized finite and affine $W$-algebras in type $A$
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 $\mathfrak{gl}_N$. The new family unifies a few known classes of $W$-algebras. In particular, for the column-partition $\lambda$ we recover the affine $W$-algebras $W^k(\mathfrak{gl}_N,f)$ of Kac, Roan and …