Whitney algebras and Grassmann's regressive products

TL;DR

This paper explores geometric products in exterior and Whitney algebras, linking them with classical operators and providing algebraic encodings that simplify proofs of key identities and exchange relations.

Contribution

It introduces algebraic encodings of tensor and Whitney algebras using letterplace and divided power polarization operators, clarifying their structure and relations.

Findings

Established connections between geometric products and classical operators.

Provided simplified proofs of exchange relations in Whitney algebras.

Linked tensor powers and Whitney algebras to letterplace algebra encodings.

Abstract

Geometric products on tensor powers $\Lambda(V)^{\otimes m}$ of an exterior algebra and on Whitney algebras \cite{crasch} provide a rigorous version of Grassmann's {\it regressive products} of 1844 \cite{gra1}. We study geometric products and their relations with other classical operators on exterior algebras, such as the Hodge $\ast-$operators and the {\it join} and {\it meet} products in Cayley-Grassmann algebras \cite{BBR, Stew}. We establish encodings of tensor powers $\Lambda(V)^{\otimes m}$ and of Whitney algebras $W^m(M)$ in terms of letterplace algebras and of their geometric products in terms of divided powers of polarization operators. We use these encodings to provide simple proofs of the Crapo and Schmitt exchange relations in Whitney algebras and of two typical classes of identities in Cayley-Grassmann algebras.