Basics of Geometric Algebra

Geometric algebras are a family of algebras defined over vector spaces so that operations on subspaces become algebraic operations. They are also known as Clifford algebras. In general a Clifford algebra can be built over any field, but "geometric algebra" conventionally means one generated by a vector space over the reals, and that is the only case considered here.

The algebra

Let be a quadratic space: a -dimensional vector space with a quadratic form . The integers , and count the basis vectors squaring to , and respectively — together they are the signature of the algebra.

The geometric algebra is the associative algebra over that quadratic space which forms a ring. Both and sit inside it as subspaces. A general element is called a multivector, and the algebra's product — written simply as juxtaposition — is the geometric product.

The property that separates geometric algebra from other associative algebras is that squaring a vector yields a scalar:

Inner and outer products

For two vectors the geometric product splits into a symmetric and an antisymmetric part, and those two parts are themselves meaningful products:

The symmetric part is the inner product , which carries the metric of the algebra. The antisymmetric part is the outer product , which spans: it builds higher-dimensional subspaces out of lower-dimensional ones. The quantity is a bivector, and it represents the plane segment spanned by the two vectors rather than a number.

This is the pairing that makes the algebra useful. One product measures, the other constructs, and both are halves of a single product.

Blades and grades

Write the basis vectors of as . Taking geometric products of distinct basis vectors in every possible combination gives the algebraic basis of , whose elements are called basis blades. There are of them. For the corresponding blade is

so for we get , written by convention. A general multivector is a linear combination of basis blades.

As a vector space the algebra is isomorphic to the exterior algebra, which makes it graded. The grade of a blade is the number of basis vectors in it, , and the grade projection keeps only the grade- part:

Scalars are then grade and vectors grade . The highest-grade blade is the pseudoscalar .

Grade projection also extends the inner and outer products from vectors to arbitrary multivectors:

Read together these say what the two products do to dimension: the inner product lowers grade, the outer product raises it.

Two further products built from the geometric product appear later — the commutator and anticommutator:

Duality and the reverse

For a non-degenerate algebra () the dual of a blade is its product with the inverse pseudoscalar,

which exchanges a subspace for its orthogonal complement. Duality is what lets the same geometric object be described in two equivalent ways, a point taken up in the conformal model.

Applying transformations needs one more operator, the reverse. For a blade it reverses the order of the factors:

A small example: the plane

The geometric algebra of the Euclidean plane makes the definitions concrete. The quadratic space is with basis vectors , both squaring to , so the algebra is with algebraic basis

Its pseudoscalar is . Squaring it:

An element squaring to has appeared in a purely real algebra — the complex numbers arrive as a consequence of the definitions rather than as an extra construction. The same happens for other familiar algebras: , , and , the quaternions.

Versors

In a non-degenerate algebra , a multivector formed as the geometric product of invertible vectors, , is called a versor. Versors are invertible, and with the geometric product they form the Clifford group. Restricted to unit norm the inverse is just the reverse, , giving the group; the subgroup with even is .

The unit vectors making up a versor can be read as hyperplanes, and by the Cartan–Dieudonné theorem any orthogonal transformation is a composition of reflections in hyperplanes. So these groups represent orthogonal transformations directly, as double covers of and .

A versor acts on any multivector by the same sandwich product:

This operation preserves grade, so it does not change what a multivector represents — a transformed line is still a line. This is the property the rest of these pages rely on: a transformation is written once and applies to every element of the algebra, not just to vectors. The extension is called an outermorphism.

Next: the conformal model, which chooses a specific signature so that translations join rotations as versors.