W-ALGEBRAS FROM SOLITON-EQUATIONS AND HEISENBERG SUBALGEBRAS

We derive sufficient conditions under which the ''second'' Hamiltonian structure of a class of generalized KdV-hierarchies defines one of th e classical W-algebras obtained through Drinfel'd-Sokolov Hamiltonian reduction. These integrable hierarchies are associated to the Heisenbe rg subalgebras of an untwisted affine Kac-Moody algebra. When the prin cipal Heisenberg subalgebra is chosen, the well-known connection betwe en the Hamiltonian structure of the generalized Drinfel'd-Sokolov hier archies-the Gel'fand-Dickey algebras-and the W-algebras associated to the Casimir invariants of a Lie algebra is recovered. After carefully discussing the relations between the embeddings of A(1) = sl(2, C) int o a simple Lie algebra g and the elements of the Heisenberg subalgebra s of g((1)), we identify the class of W-algebras that can be defined i n this way. For A,, this class only includes those associated to the e mbeddings labelled by partitions of the form n + 1 = k(m)) + q(1) and n + 1 = k(m + 1) + k(m) + q(1). (C) 1995 Academic Press, Inc.