Exterior Algebra and Grassmann Algebra

Abstract In this chapter, exterior algebras and Grassmann algebras are discussed. In particular, their morphisms, inherited from the tensor algebra, are presented in a general framework. In some texts, the terms exterior algebra and Grassmann algebra are considered to be synonyms; however, this is not the case here! In this chapter, the differences between the exterior algebra and the Grassmann algebra are presented as clearly as possible, in order to avoid confusion later on. The quasi-Hodge isomorphism is thus presented and studied. In addition, based on Grassmann’s use of the Hodge star operator to define the regressive product, the quasi-Hodge operator is used to obtain the quasi-Hodge isomorphism, with the aid of a metric structure. At the end of the chapter, the Hodge isomorphisms are examined.