ON MATRIX KP AND SUPER-KP HIERARCHIES IN THE HOMOGENEOUS GRADING

Abstract

Constrained KP and super-KP hierarchies of integrable equations (generalized NLS hierarchies) are systematically produced through a Lie-algebraic AKS matrix framework associated with the homogeneous grading. The role played by different regular elements in defining the corresponding hierarchies is analyzed, as well as the symmetry properties under the Weyl group transformations. The coset structure of higher order Hamiltonian densities is proven. For a generic Lie algebra the hierarchies considered here are integrable and essentially dependent on continuous free parameters. The bosonic hierarchies studied in Refs. 1 and 2 are obtained as special limit restrictions on Hermitian symmetric spaces. In the supersymmetric case the homogeneous grading is introduced consistently by using alternating sums of bosons and fermions in the spectral parameter power series. The bosonic hierarchies obtained from [Formula: see text] and the supersymmetric ones derived from the N=1 affinization of sl (2), sl (3) and osp (1|2) are explicitly constructed. An unexpected result is found: only a restricted subclass of the sl (3) bosonic hierarchies can be supersymmetrically extended while preserving integrability.