Mathematics

In 1985 Zamolodchikov constructed a "non-linear" extension of the Virasoro algebra known as W_3-algebra. This is the one of the first appearance of a rich class of algebraic structures, known as W-algebras, which are intimately related to physical theories with symmetry and revealed many applications in mathematics. In the first part of the talk I will review some facts about the general theory of W-algebras. Then, I will explain how to describe quantum finite and classical affine W-algebras using Lax operators. In the quantum finite case this operator satisfies a generalized Yangian identity, while in the classical affine case it is used to construct integrable hierarchies of Hamiltonian equations in Lax form. This is a joint work with A. De Sole and V.G. Kac.