Abstract
We explicitly demonstrate with a help of a computer that Clifford algebra Cℓ(B) of a bilinear form B with a non-trivial antisymmetric part A is isomorphic as an associative algebra to the Clifford algebra Cℓ(Q) of the quadratic form Q induced by the symmetric part of B [in characteristic ≠ 2], However, the multivector structure of Cℓ(B) depends on A and is therefore different than the one of Cℓ(Q). Operation of reversion is still an anti-automorphism of Cℓ(B). It preserves a new kind of gradation in ⋀ V determined by A but it does not preserve the gradation in ⋀ V. The demonstration is given for Clifford algebras in real and complex vector spaces of dimension ≤ 9 with a help of a Maple package ‘Clifford’. The package has been developed by one of the authors to facilitate computations in Clifford algebras of an arbitrary bilinear form B.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
R. Abłamowicz: ‘Clifford algebra computations with Maple’, Proceedings of CAP Summer School in Theoretical Physics, “Geometric (Clifford) Algebras in Physics,” Banff, 1995 (to appear). Technical Report No. 1995-1, Department of Mathematics, Gannon University, October 1995.
R. Abłamowicz: 1996, ‘Clifford’ — Maple V package for Clifford algebra computations, ver. 2, available at: http://www.gannon.edu/service/dept/mathdept.
C. Chevalley: 1954, ‘The Algebraic Theory of Spinors’, Columbia University Press, New York.
A. Crumeyrolle: 1990, ‘Orthogonal and Symplectic Clifford Algebras: Spinor Structures’, Kluwer, Dordrecht.
J. Helmstetter: 1982, ‘Algebres de Clifford et algebres de Weyl’, Cahiers Math. 25, Montpellier.
P. Lounesto, R. Mikkola, and V. Vierros: 1987, ‘CLICAL User Manual’, Helsinki University of Technology, Institute of Mathematics, Research Reports A248, Helsinki.
P. Lounesto: 1993, ‘What is a bivector?’, in ‘Spinors, Twistors, Clifford Algebras and Quantum Deformations’, Proceedings of the Second Max Born Seminar Series, Wroclaw, Poland, 1992; eds. Z. Oziewicz, B. Jancewicz, and A. Borowiec Kluwer, Dordrecht, pp. 153-158.
P. Lounesto: 1995, ‘Crumeyrolle’s bivectors and spinors’, pp. 137–166 in R. Abłamowicz, P. Lounesto (eds.): Clifford Algebras and Spinor Structures, A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919–1992). Kluwer, Dordrecht, 1995. ‘Maple V Release 3 for DOS and Windows,’ 1994, Waterloo Maple Software, Waterloo, Ontario.
Z. Oziewicz: 1986, ‘From Grassmann to Clifford’, pp. 245–255 in J.S.R. Chisholm, A.K. Common (eds.): Clifford Algebras and their Applications in Mathematical Physics (Canterbury, 1985). Reidel, Dordrecht, 1986.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Birkhäuser Boston
About this chapter
Cite this chapter
Abłamowicz, R., Lounesto, P. (1996). On Clifford Algebras of a Bilinear Form with an Antisymmetric Part. In: Abłamowicz, R., Parra, J.M., Lounesto, P. (eds) Clifford Algebras with Numeric and Symbolic Computations. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4615-8157-4_11
Download citation
DOI: https://doi.org/10.1007/978-1-4615-8157-4_11
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4615-8159-8
Online ISBN: 978-1-4615-8157-4
eBook Packages: Springer Book Archive