Abstract
Typically, Geometric Algebra (GA) is introduced via choosing an orthogonal basis, and defining how multiplication acts on this basis according to some simple rules. This works well computationally, but can obscure insight mathematically. In particular, operations defined in terms of coordinates on a multivector basis can be difficult to rigorously show to be “coordinate-free”, especially in large algebras. This paper explores the use of the “universal property” to ensure that operations are “coordinate-free” by construction. To build some insight for applying the universal property, we draw parallels to the process of writing recursive programs. We then demonstrate a novel result using this approach by deriving a universal property of the even subalgebra. Armed with this second universal property, we provide an explicit construction for a well-known equivalence between any Clifford algebra and its “one-up” even subalgebra. We conclude with some remarks about formalization of these ideas in a theorem prover.
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Notes
- 1.
In this paper, we will use “Clifford algebra” to refer to this object, and “geometric algebra” to refer to the field of study.
- 2.
So-named in reference to the mathematician Haskell Curry.
- 3.
Note we call this a definition as we are defining \({\text {lift}}^{+}\), which has computational content just like the recursor for lists did.
- 4.
- 5.
for which [6] proves only existence.
- 6.
Confusingly, [3] uses \(u\llcorner U\) as notation for \(u \rfloor U\), with the symbol flipped.
- 7.
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Acknowledgements
The authors would like to thank David Cohoe for illuminating the ideas behind (10) that led to the rest of this paper. The first author is funded by a scholarship from the Cambridge Trust.
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Wieser, E., Lasenby, J. (2024). Computing with the Universal Properties of the Clifford Algebra and the Even Subalgebra. 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_17
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