Reworking on affine exterior algebra of Grassmann, Peano and his school

TL;DR

This paper explores the historical development and mathematical construction of affine exterior algebra over affine spaces, highlighting Peano's innovative definitions and analyzing metric versus affine approaches, with applications in geometry and mechanics.

Contribution

It revisits and clarifies the construction of affine exterior algebra, emphasizing Peano's unique definition methods and the role of metric concepts, filling a gap in the mathematical literature.

Findings

Peano's definition by abstraction is a significant contribution.

Metric concepts can often be eliminated in affine exterior algebra.

Applications in geometry and mechanics illustrate the algebra's usefulness.

Abstract

In this paper a construction of affine exterior algebra of Grassmann, with a special attention to the revisitation of this subject operated by Peano and his School, is examined from a historical viewpoint. Even if the exterior algebra over a vector space is a well known concept, the construction of an exterior algebra over an affine space, in which points and vectors coexist, has been neglected. This paper wants to fill this lack. Some attention is given to the introduction of defining by abstraction (today called definition by quotienting or by equivalence relation), a procedure due to and used by Peano to define geometric forms, basic elements of an affine exterior algebra. This Peano's innovative way of defining, is a relevant contribution to mathematics. It is observed that in the construction of an affine exterior algebra on the Euclidean three-dimensional space, Grassmann and Peano make use of metric concepts: an accurate analysis shows that, in some cases, the metric aspects can be eliminated, putting into evidence the sufficiency of the underlying affine structure of the Euclidean space. In the final part of the paper some geometrical and mechanical applications and interpretations of the affine exterior algebra given by Grassmann and Peano are presented.