Grassmann Geometric Calculus and Supersymmetric Algebras

Capelli's technique of VIRTUAL VARIABLES finds its natural setting in the context of supersymmetric algebras and Lie superalgerba actions, where the variables are allowed to have a different signature from the signature of the variables one starts with.

Our purpose is to propose an extension of the method to the Grassmann Geometric Calculus (CG-Algebra Theory). We introduce a superalgebraic module which can be endowed by a pair of associative products, called the JOIN and the MEET. The submodule spanned by the NEGATIVE variables is essentially the usual EXTERIOR CG-Algebra, that spanned by the POSITIVE variables is its SYMMETRIC counterpart. The process of COMPLETE NEGATIVE POLARIZATION gives the connection between these two notable sub(double) algebras. The present approach allows us to develop a new method to deal with covariants of skew-symmetric tensors, as well as a new SYMBOLIC INVARIANT notation to manage geometric calculi.