Computing with the Universal Properties of the Clifford Algebra and the Even Subalgebra

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.