Mathematics

We will give the definition of the classical W-algebras W(g,f),  where g is a Lie algebra and f in g is a nilpotent element, in the framework of Poisson vertex algebras. We will discuss some sufficient conditions on the nilpotent element f which guarantee the existence of generalized Drinfeld-Sokolov hierarchies for classical W-algebras. In the case of g=gl_N, generalized Drinfeld-Sokolov hierarchies are constructed for arbitrary nilpotent elements using the notion of Adler type pseudodifferential operators.