Clifford algebra

Let $V$ be a vector space over a field $k$ , and $Q:V\times V\to k$ a symmetric bilinear form. Then the Clifford algebra $\operatorname{Cliff}(Q,V)$ is the quotient of the tensor algebra $\mathcal{T}(V)$ by the relations

$v\otimes w+w\otimes v=-2Q(v,w)\qquad\forall v,w\in V.$

Since the above relationship is not homogeneous in the usual $\mathbb{Z}$ -grading on $\mathcal{T}(V)$ , $\operatorname{Cliff}(Q,V)$ does not inherit a $\mathbb{Z}$ -grading. However, by reducing mod 2, we also have a $\mathbb{Z}_{2}$ -grading on $\mathcal{T}(V)$ , and the relations above are homogeneous with respect to this, so $\operatorname{Cliff}(Q,V)$ has a natural $\mathbb{Z}_{2}$ -grading, which makes it into a superalgebra.

In addition, we do have a filtration on $\operatorname{Cliff}(Q,V)$ (making it a filtered algebra), and the associated graded algebra $\operatorname{Gr}\operatorname{Cliff}(Q,V)$ is simply $\Lambda^{*}V$ , the exterior algebra of $V$ . In particular,

$\dim\operatorname{Cliff}(Q,V)=\dim\Lambda^{*}V=2^{\dim V}.$

The most commonly used Clifford algebra is the case $V=\mathbb{R}^{n}$ , and $Q$ is the standard inner product with orthonormal basis $e_{1},\ldots,e_{n}$ . In this case, the algebra is generated by $e_{1},\ldots,e_{n}$ and the identity of the algebra $1$ , with the relations

	$\displaystyle e_{i}^{2}$	$\displaystyle=-1$
	$\displaystyle e_{i}e_{j}$	$\displaystyle=-e_{j}e_{i}\quad(i\neq j)$

Trivially, $\operatorname{Cliff}(\mathbb{R}^{0})=\mathbb{R}$ , and it can be seen from the relations above that $\operatorname{Cliff}(\mathbb{R})\cong\mathbb{C}$ , the complex numbers, and $\operatorname{Cliff}(\mathbb{R}^{2})\cong\mathbb{H}$ , the quaternions.

On the other ha nd, for $V=\mathbb{C}^{n}$ we get the particularly answer of

$\operatorname{Cliff}(\mathbb{C}^{2k})\cong\mathrm{M}_{2^{k}}(\mathbb{C})\qquad% \operatorname{Cliff}(\mathbb{C}^{2k+1})=\mathrm{M}_{2^{k}}(\mathbb{C})\oplus% \mathbf{M}_{2^{k}}(\mathbb{C}).$

Title	Clifford algebra
Canonical name	CliffordAlgebra
Date of creation	2013-03-22 13:18:05
Last modified on	2013-03-22 13:18:05
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	9
Author	rmilson (146)
Entry type	Definition
Classification	msc 15A66
Classification	msc 11E88