Affine Lie Algebraic Origin of Constrained KP Hierarchies

We present an affine $sl (n+1)$ algebraic construction of the basic constrained KP hierarchy. This hierarchy is analyzed using two approaches, namely linear matrix eigenvalue problem on hermitian symmetric space and constrained KP Lax formulation and we show that these approaches are equivalent. The model is recognized to be the generalized non-linear Schr\"{o}dinger ($\GNLS$) hierarchy and it is used as a building block for a new class of constrained KP hierarchies. These constrained KP hierarchies are connected via similarity-B\"{a}cklund transformations and interpolate between $\GNLS$ and multi-boson KP-Toda hierarchies. Our construction uncovers origin of the Toda lattice structure behind the latter hierarchy.