Structure of classical (finite and affine) $\mathcal{W}$-algebras

Abstract

First, we derive an explicit formula for the Poisson bracket of the classical finite $\mathcal{W}$-algebra ${\mathcal{W}}^{\mathrm{fin}}(\mathfrak{g},f)$, the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra $\mathfrak{g}$ and its nilpotent element $f$. On the other hand, we produce an explicit set of generators and we derive an explicit formula for the Poisson vertex algebra structure of the classical affine $\mathcal{W}$-algebra $\mathcal{W}(\mathfrak{g},f)$. As an immediate consequence, we obtain a Poisson algebra isomorphism between ${\mathcal{W}}^{\mathrm{fin}}(\mathfrak{g},f)$ and the Zhu algebra of $\mathcal{W}(\mathfrak{g},f)$. We also study the generalized Miura map for classical $\mathcal{W}$-algebras.