Grade free product formulae from Grassmann Hopf gebras

TL;DR

This paper introduces general, grade-free formulae for various products in Clifford algebras derived from Grassmann Hopf algebras, applicable to any grade, form, and multivector, improving computational efficiency and generality.

Contribution

It provides a unified, grade-independent approach to Clifford product formulas using Grassmann Hopf algebra, extending known formulas and enabling applications to non-symmetric and degenerated forms.

Findings

Derived general product formulae valid for any grade and multivector

Unified approach includes meet, join, contraction, and Clifford products

Applicable to symplectic Clifford algebras and non-symmetric forms

Abstract

In the traditional approaches to Clifford algebras, the Clifford product is evaluated by recursive application of the product of a one-vector (span of the generators) on homogeneous i.e. sums of decomposable (Grassmann), multivectors and later extended by bilinearity. The Hestenesian 'dot' product, extending the one-vector scalar product, is even worse having exceptions for scalars and the need for applying grade operators at various times. Moreover, the multivector grade is not a generic Clifford algebra concept. The situation becomes even worse in geometric applications if a meet, join or contractions have to be calculated. Starting from a naturally graded Grassmann Hopf gebra, we derive general formulae for the products: meet and join, comeet and cojoin, left/right contraction, left/right cocontraction, Clifford and co-Clifford products. All these product formulae are valid for any grade and any inhomogeneous multivector factors in Clifford algebras of any bilinear form, including non-symmetric and degenerated forms. We derive the three well known Chevalley formulae as a specialization of our approach and will display co-Chevalley formulae. The Rota--Stein cliffordization is shown to be the generalization of Chevalley deformation. Our product formulae are based on invariant theory and are not tied to representations/matrices and are highly computationally effective. The method is applicable to symplectic Clifford algebras too.