Poisson structures of deformed $W$-algebras and classical $W$-algebras

In this paper, we explain how classical $W$-algebras can be obtained as limits of $q$-deformed $W$-algebras in type $A$. To this end, we give a detailed examination of the construction of deformed $W$-algebras via Hamiltonian reduction of deformed affine Poisson algebras. We first derive an affine Poisson vertex algebra (PVA) from the Poisson structure of the deformed affine Poisson algebra. This construction identifies a specific set of generators of the deformed $W$-algebra with generators of the classical $W$-algebra.