We analyze the W_N^l algebras according to their conjectured realization asthe second Hamiltonian structure of the integrable hierarchy resulting from theinterchange of x and t in the l^{th} flow of the sl(N) KdV hierarchy. The W_4^3algebra is derived explicitly along these lines, thus providing further supportfor the conjecture. This algebra is found to be equivalent to that obtained bythe method of Hamiltonian reduction. Furthermore, its twisted versionreproduces the algebra associated to a certain non-principal embedding of sl(2)into sl(4), or equivalently, the u(2) quasi-superconformal algebra. The generalaspects of the W_N^l algebras are also presented.
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