W-algebras in type A and the Arakawa-Moreau conjecture

W-algebras are an important class of vertex algebras associated with a reductive Lie algebra g, a nilpotent element f ∈ g and a scalar k ∈ C, which are closely related with various area of mathematics such as integrable systems, two-dimensional conformal field theories, modular representation theory, four dimensional gauge theory, and geometric Langlands program. Moreover, there has been a renewed interest in W-algebras since they appear as invariants of Argyres-Douglas theory via the 4D/2D correspondence recently discovered in physics. However, despite of the importance of W-algebras the problem of finding all the generators for every affine W-algebra remains unsolved. The only results known so far are from Kac-Wakimoto for minimal nilpotent elements, and from Arakawa-Molev for rectangular nilpotent elements, with the restriction of g = glN. In this thesis we obtained an explicit list of generators of W-algebras of type A associated with quasi-rectangular nilpotent elements. This is a nice generalization of the aforementioned results, since both are quasi-rectangular. Furthermore, as an application we were able to confirm a conjecture of Anne Moreau and Tomoyuki Arakawa in some cases on the isomorphism of simple quotients of W-algebras. This is a promising result since it confirms also some expectations by physicists that arose in the recent study of the 4D/2D correspondence.