Generalized Grassmann Algebras and its Connection to the Extended Supersymmetric Models

Abstract

It is shown that the fermionic Heisenberg-Weyl algebra with 2N=D fermionic generators is equivalent to the generalized Grassmann algebra with two fractional generators. The 2,3 and 4 dimensional Heisenberg - Weyl algebra is explicitly given in terms of the fractional generators. These algebras are used for the formulation of the N=2,3,4 extended supersymmetry. As an example we reformulate the Lax approach of the supersymmetric Korteweg - de Vries equation in terms of the generators of the generalized Grassmann algebra.