Grassmann coordinates and Lie algebras for unified models

Abstract

A variety of Lie algebras and certain classes of representations can be constructed using Grassmann variables regarded as Lorentz scalar coordinates belonging to an internal space. The generators are realized as combinations of multilinear products of the coordinates and derivative operators, while the representations emerge as antisymmetric polynomials in the variables and are thus severely restricted. The nature of these realizations and the interconnections between various subalgebras, for N independent complex anticommuting coordinates, is explored. The addition of such Grassmann coordinates to the usual spacetime manifold provides a natural superfield setting for a unified theory of symmetries of elementary particles. The particle content can be further restricted by imposing discrete symmetries (Lie algebra automorphisms). For the case N=5 some anomaly free choices of multiplets are derived through the imposition of specific superfield duality conditions.