Formula omitted]-Algebras from Soliton Equations and Heisenberg Subalgebras

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We derive sufficient conditions under which the "second" Hamiltonian structure of a class of generalized KdV-hierarchies defines one of the classical W-algebras obtained through Drinfel′d-Sokolov Hamiltonian reduction. These integrable hierarchies are associated to the Heisenberg subalgebras of an untwisted affine Kac-Moody algebra. When the principal Heisenberg subalgebra is chosen, the well-known connection between the Hamiltonian structure of the generalized Drinfel′d-Sokolov hierarchies-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 A1 = sl(2, C) into a simple Lie algebra g and the elements of the Heisenberg subalgebras of g(1), we identify the class of W-algebras that can be defined in this way. For An, this class only includes those associated to the embeddings labelled by partitions of the form n + 1 = k(m) + q(1) and n + 1 = k(m + 1) + k(m) + q(1).