This thesis presents the structure of classical and quantum W-algebras for type A. The classical W-algebra originates from the Hamiltonian reduction of the classical affine Poisson vertex algebra, first studied in the context of integrable systems and later extended to a broader family with explicitly described Poisson structures. The quantum W-algebra, defined via the BRST complex, was shown by De Sole and Kac to be a quantization of the classical one, since the quasi-classical limit of the BRST complex induces the classical W-algebra.

In this thesis, we construct weak generators of the classical W-algebra for type A based on explicit Poisson brackets. Here, a weak generating set refers to a subset of elements that generate the entire algebra through Poisson brackets. Through the correspondence between the classical and quantum cases, these weak generators are lifted to the quantum setting, leading to a weak generation property of the quantum W-algebra, analogous to the strong generation property established by Kac and Wakimoto.