The Grassmann Algebra and its Differential Identities

Abstract

Let G be the infinite dimensional Grassmann algebra over an infinite field F of characteristic different from two. In this paper we study the differential identities of G with respect to the action of a finite dimensional Lie algebra L of inner derivations. We explicitly determine a set of generators of the ideal of differential identities of G. Also in case F is of characteristic zero, we study the space of multilinear differential identities in n variables as a module for the symmetric group Sn and we compute the decomposition of the corresponding character into irreducibles. Finally, we prove that unlike the ordinary case the variety of differential algebras with L action generated by G has no almost polynomial growth.