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On Clifford Algebras of a Bilinear Form with an Antisymmetric Part

In: Clifford Algebras with Numeric and Symbolic Computations

Author

Listed:
  • Rafał Abłamowicz

    (Gannon University, Department of Mathematics)

  • Pertti Lounesto

    (Helsinki University of Technology, Institute of Mathematics)

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.

Suggested Citation

  • Rafał Abłamowicz & Pertti Lounesto, 1996. "On Clifford Algebras of a Bilinear Form with an Antisymmetric Part," Springer Books, in: Rafał Abłamowicz & Josep M. Parra & Pertti Lounesto (ed.), Clifford Algebras with Numeric and Symbolic Computations, pages 167-188, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4615-8157-4_11
    DOI: 10.1007/978-1-4615-8157-4_11
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