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

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.Comment: 36 pages, LaTe

On the Octonionic M-superalgebra

The generalized supersymmetries admitting abelian bosonic tensorial central charges are classified in accordance with their division algebra structure (over ${\bf R}$, ${\bf C}$, ${\bf H}$ or ${\bf O}$). It is shown in particular that in D=11 dimensions, the $M$-superalgebra admits a consistent octonionic formulation, involving 52 real bosonic generators (in place of the 528 of the standard $M$-superalgebra). The octonionic $M5$ (super-5-brane) sector coincides with the octonionic $M1$ and $M2$ sectors, while in the standard formulation these sectors are all independent. The octonionic conformal and superconformal $M$-algebras are explicitly constructed. They are respectively given by the $Sp(8|{\bf O})$ ($OSp(1,8|{\bf O})$) (super)algebra of octonionic-valued (super)matrices, whose bosonic subalgebra consists of 232 (and respectively 239) generators.Comment: 17 pages. Proceedings of the Workshop on Integrable Theories, Solitons and Duality, S. Paulo, July 2002. In JHE

N=1,2 Super-NLS Hierarchies as Super-KP Coset Reductions

We define consistent finite-superfields reductions of the $N=1,2$ super-KP hierarchies via the coset approach we already developped for reducing the bosonic KP-hierarchy (generating e.g. the NLS hierarchy from the $sl(2)/U(1)-{\cal KM}$ coset). We work in a manifestly supersymmetric framework and illustrate our method by treating explicitly the $N=1,2$ super-NLS hierarchies. W.r.t. the bosonic case the ordinary covariant derivative is now replaced by a spinorial one containing a spin ${\textstyle {1\over 2}}$ superfield. Each coset reduction is associated to a rational super-\cw algebra encoding a non-linear super-\cw_\infty algebra structure. In the $N=2$ case two conjugate sets of superLax operators, equations of motion and infinite hamiltonians in involution are derived. Modified hierarchies are obtained from the original ones via free-fields mappings (just as a m-NLS equation arises by representing the $sl(2)-{\cal KM}$ algebra through the classical Wakimoto free-fields).Comment: 27 pages, LaTex, Preprint ENSLAPP-L-467/9

Division Algebras and Extended SuperKdVs

The division algebras R, C, H, O are used to construct and analyze the N=1,2,4,8 supersymmetric extensions of the KdV hamiltonian equation. In particular a global N=8 super-KdV system is introduced and shown to admit a Poisson bracket structure given by the "Non-Associative N=8 Superconformal Algebra".Comment: 6 pages, LaTex; Talk given at the XXXVII Karpacz Winter School in Theoretical Physics (February 2001). To appear in the proceeding