On Matrix KP and Super-KP Hierarchies in the Homogeneous Grading

TL;DR

This paper systematically constructs and analyzes matrix KP and super-KP hierarchies using Lie algebraic methods, revealing new insights into their structure, symmetries, and supersymmetric extensions.

Contribution

It introduces a Lie algebraic AKS-matrix framework for deriving hierarchies and explores their symmetry properties and supersymmetric extensions, including explicit constructions for specific algebras.

Findings

Hierarchies depend on continuous free parameters.

Supersymmetric extensions are limited for certain hierarchies.

Coset structures of Hamiltonian densities are established.

Abstract

Constrained KP and super-KP hierarchies of integrable equations (generalized NLS hierarchies) are systematically produced through a Lie algebraic AKS-matrix framework associated to the homogeneous grading. The role played by different regular elements to define 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.\par For a generic Lie algebra the hierarchies here considered are integrable and essentially dependent on continuous free parameters. The bosonic hierarchies studied in \cite{{FK},{AGZ}} are obtained as special limit restrictions on hermitian symmetric-spaces.\par In the supersymmetric case the homogeneous grading is introduced consistently by using alternating sums of bosons and fermions in the spectral parameter power series.\par The bosonic hierarchies obtained from ${\hat {sl(3)}}$ and the supersymmetric ones derived from the $N=1$ affinization of $sl(2)$, $sl(3)$ and $osp(1|2)$ are explicitly constructed. \par An unexpected result is found: only a restricted subclass of the $sl(3)$ bosonic hierarchies can be supersymmetrically extended while preserving integrability.