Clifford algebras are associative algebras constructed from quadratic forms on vector spaces. They can be viewed as generalizations of the real numbers, complex numbers, and quaternions. These algebras have applications in geometry and theoretical physics.

Given an $F$ vector space $V$ with a quadratic form $Q:V\rightarrow F$, the Clifford algebra $C\ell(V,Q)$ is a quotient of the tensor algebra of $V$ by the ideal generated by elements of the form $v\otimes v - Q(v)\cdot 1$ for $v\in V$.

The algebras are eponymous for William Kingdon Clifford's work with them at the end of his life, although he himself called them "geometric algebras." As the name suggests, these algebras have applications in geometry.


  1. When $Q=0$, the algebra $C\ell(V,Q)$ is the exterior algebra of $V$.

  2. When $F=\Bbb R$ and $V=\Bbb R^2$ and $Q$ has the signature $(1,-1)$, the algebra $C\ell(V,Q)\cong \Bbb C$ as $\Bbb R$ algebras.

  3. When $F=\Bbb R$ and $V=\Bbb R^4$ and $Q$ has the signature $(1,-1,-1,-1)$, the algebra $C\ell(V,Q)\cong \Bbb H$ as $\Bbb R$ algebras.

Clifford algebras over $\Bbb R$ and $\Bbb C$ have been used in recent decades for theoretical physics. Advocates of so-called geometric algebra interpret elements of the algebra geometrically as vectors and subspaces of $V$. This interpretation allows convenient execution of tasks like projection, reflection and rotation to be done with the algebra product.

Given an ordered basis $\{b_i\mid i\in 1\dots n \}$ for finite dimensional $V$, a standard basis for $C\ell(V,Q)$ is given by forming all possible products of basis elements such that the indices appearing are strictly ascending from left to right. Thus $b_1\otimes b_3\otimes b_4$ is a basis element, but $b_1\otimes b_4\otimes b_3$ is not. The empty product of basis elements is taken to be the identity of the algebra. If $V$ is $n$ dimensional, then $C\ell(V,Q)$ will be $2^n$ dimensional.

With the standard basis, the Clifford algebra has a natural grading. The grade-$k$ subspace is the subspace generated by all elements of the standard basis which are products of exactly $k$ of the $b_i$. Grade $k$ elements are called k-vectors or k-blades, and are interpreted by geometers as $k$-volume elements.

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