Geometric Algebra - Definition and Notation

Definition and Notation

Given a finite dimensional real quadratic space V = Rn with quadratic form Q : V → R, the geometric algebra for this quadratic space is the Clifford algebra Cℓ(V,Q).

The algebra product is called the geometric product. It is standard to denote the geometric product by juxtaposition.

For quadratic forms of any signature, an orthogonal basis {e₁,...,e_n} can be found for V such that each e_i2 is either −1, 0 or +1. The number of e_i's associated with each of these three values is expressed by the signature, which is an invariant of the quadratic form.

When Q is nondegenerate there are no 0's in the signature, and so an orthogonal basis of V exists with p elements squaring to 1 and q elements squaring to −1, with p + q = n. We denote this algebra . For example, models 3D Euclidean space, relativistic spacetime and a 3D Conformal Geometric algebra.