Grade Free Product Formulæ from Grassmann-Hopf Gebras

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.