Classical \(W\)-algebras for centralizers

A. I. Molev and E. Ragoucy

Abstract

We introduce a new family of Poisson vertex algebras \(\mathcal{W}(\mathfrak{a})\) analogous to the classical \(\mathcal{W}\)-algebras. The algebra \(\mathcal{W}(\mathfrak{a})\) is associated with the centralizer \(\mathfrak{a}\) of an arbitrary nilpotent element in \(\mathfrak{gl}_N\). We show that \(\mathcal{W}(\mathfrak{a})\) is an algebra of polynomials in infinitely many variables and produce its free generators in an explicit form. This implies that \(\mathcal{W}(\mathfrak{a})\) is isomorphic to the center at the critical level of the affine vertex algebra associated with \(\mathfrak{a}\).