Quasideterminants and differential realizations of W-algebras

Abstract

In this work, we study the Hamiltonian reduction, used in constructing finite rectangular $W$-algebras, of the big-cell differential realization of ${\mathfrak{gl}}_{np}$. We establish an intrinsic factorization of the reduced differential realization, identifying its Yangian action with the coproduct action on a tensor product of $n$ evaluation differential realizations of ${\mathfrak{gl}}_p$. Corner quasideterminants provide the essential noncommutative mechanism through which Hamiltonian reduction transforms the higher-rank differential realization into a factorized lower-rank Yangian action.