2004 | OriginalPaper | Chapter
Grade Free Product Formulæ from Grassmann-Hopf Gebras
Author : Bertfried Fauser
Published in: Clifford Algebras
Publisher: Birkhäuser Boston
Included in: Professional Book Archive
Activate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by
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), multi-vectors 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 formulæ for the products: meet and join, comeet and cojoin, left/right contraction, left/right cocontraction, Clifford and co-Clifford products. All these product formulæ 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 formulæ as a specialization of our approach and will display co-Chevalley formulæ. The Rota—Stein cliffordization is shown to be the generalization of Chevalley deformation. Our product formulæ 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.