N ov 1 99 2 GENERALIZED DRINFELD - SOKOLOV HIERARCHIES AND W - ALGEBRAS *

We review the construction of Drinfeld-Sokolov type hierarchies and classical W-algebras in a Hamiltonian symmetry reduction framework. We describe the list of graded regular elements in the Heisenberg subalgebras of the nontwisted loop algebra ℓ(gl n) and deal with the associated hierarchies. We exhibit an sl 2 embedding for each reduction of a Kac-Moody Poisson bracket algebra to a W-algebra of gauge invariant differential polynomials. In this talk I wish to describe some recent results on the construction of KdV type hierarchies and classical W-algebras. (Proofs and further details can be found in [1], [2].) First I review the relevant aspects of the Drinfeld-Sokolov (DS) construction of KdV type hierarchies [3] and the corresponding W-algebras concentrating on the simplest case. I shall raise some questions concerning the possible generalizations, which will be (partially) answered later in the talk. As explained in detail in [1], the DS construction can be naturally understood in the framework of the Hamiltonian Adler-Kostant-Symes approach to integrable systems (e.g. [4]). The hierarchy results from a local symmetry reduction of the commuting family of Hamiltonian systems generated by the ad *-invariant Hamiltonians on the dual A * of a Lie algebra A of the form A = ℓ(G) := G ⊗ C[λ, λ −1 ], (1.1) where G itself is a centrally extended loop algebra. The space A * carries the family of compatible R Lie-Poisson brackets induced by the classical r-matrices R k ∈ End(A) given by R k := (P + − P −) • λ k , where P ± ∈ End(A) project onto the subalgebras A ± containing positive and negative powers of the spectral parameter λ, respectively, (see [5]). For simplicity, let us concentrate on the case when G = gl n ∧ , the central extension of the algebra of smooth loops in 0 dx trX ′ (x)Y (x) .