Classical W-algebras within the theory of Poisson vertex algebras

We review the Poisson vertex algebra theory approach to classical W-algebras. First, we provide a description of the Drinfeld-Sokolov Hamiltonian reduction for the construction of classical W-algebras within the framework of Poisson vertex algebras and we establish, under certain sufficient conditions, the applicability of the Lenard-Magri scheme of integrability and the existence of the corresponding integrable hierarchy of bi-Hamiltonian equations. Then we provide a Poisson vertex algebra analogue of the Gelfand-Dickey construction of classical W-algebras and we show the relations with the Drinfeld-Sokolov Hamiltonian reduction. It will be also shown that classical W-algebras are the Poisson vertex algebras which are of interest from the conformal field theory point of view.