# 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 CliffordAlgebra 2013-03-22 13:18:05 2013-03-22 13:18:05 rmilson (146) rmilson (146) 9 rmilson (146) Definition msc 15A66 msc 11E88