Structure of classical (finite and affine) \(\mathcal W\)-algebras (Q303825)

Summary: 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.