## 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 $$\mathfrak{a}$$.

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 Tuesday, November 26, 2019