  We discuss some algebraic aspects of the integrable vector non-linear
Schr\"{o}dinger hierarchies (GNLS$_{r}$). These are hierarchies of
zero-curvature equations constructed from affine Kac-Moody algebras
$\hat{sl}_{r+1}$. Using the dressing transformation method and the tau-function
formalism, we construct the N-soliton solutions of the GNLS$_{r}$ systems. The
explicit matrix elements in the case of GNLS$_{1}$ are computed using level one
vertex operator representations.
