Abstract
A (Clifford) geometric algebra is usually defined in terms of a quadratic form. A null vector v is an algebraic quantity with the property that \(v^2=0\). The universal algebra generated by taking the sums and products of null vectors over the real or complex numbers is denoted by \(\mathcal{N}\). The rules of addition and multiplication are taken to be the familiar rules of addition and multiplication of real or complex square matrices. In a series of ten definitions, the concepts of a Grassmann algebra, its dual Grassmann algebra, the associated real and complex geometric algebras, and their isomorphic real or complex coordinate matrix algebras are set down. This is followed by a discussion of affine transformations, the horosphere and conformal transformations on pseudoeuclidean spaces.
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Notes
- 1.
More general fields \(\mathbbm {F}\) can be considered as long as characteristic \(\mathbbm {F}\not =2\). For an interesting discussion of this issue see [5].
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Acknowledgements
The seeds of this note were planted almost 40 years ago in discussions with Professor Zbigniew Oziewicz, a distinguished colleague, about the fundamental role played by duality in its many different guises in mathematics and physics [14]. The discussion resurfaced recently in an exchange of emails with Information Scientist Dr. Manfred von Willich. I hope that my treatment here will open up the discussion to the wider scientific community. The author thanks the Zbigniew Oziewicz Seminar on Fundamental Problems in Physics group for many fruitful discussions of the ideas herein, which greatly helped sharpen their exposition, [15]. He also thanks the ICACGA Reviewers of this paper for helpful suggestions leading to its improvement.
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Sobczyk, G. (2024). Geometric Algebras of Compatible Null Vectors. In: Silva, D.W., Hitzer, E., Hildenbrand, D. (eds) Advanced Computational Applications of Geometric Algebra. ICACGA 2022. Lecture Notes in Computer Science, vol 13771. Springer, Cham. https://doi.org/10.1007/978-3-031-34031-4_4
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