A characterization of the natural grading of the Grassmann algebra and its non-homogeneous [formula omitted]-gradings

Abstract

Let F be any field of characteristic different from two and let E be the Grassmann algebra of an infinite dimensional F-vector space L. In this paper we will provide a condition for a Z2-grading on E to behave like the natural Z2-grading Ecan. More specifically, our aim is to prove the validity of a weak version of a conjecture presented in [10]. The conjecture poses that every Z2-grading on E has at least one non-zero homogeneous element of L. As a consequence, we obtain a characterization of Ecan by means of its Z2-graded polynomial identities. Furthermore we construct a Z2-grading on E that gives a negative answer to the conjecture.