# 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^{} $\mathrm{Cliff}(Q,V)$ is
the quotient of the tensor algebra $\mathcal{T}(V)$ by the relations

$$v\otimes w+w\otimes v=-2Q(v,w)\mathit{\hspace{1em}\hspace{1em}}\forall v,w\in V.$$ |

Since the above relationship is not homogeneous^{} in the usual
$\mathbb{Z}$-grading on $\mathcal{T}(V)$, $\mathrm{Cliff}(Q,V)$ 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 $\mathrm{Cliff}(Q,V)$ has a natural ${\mathbb{Z}}_{2}$-grading,
which makes it into a superalgebra.

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

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

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

${e}_{i}^{2}$ | $=-1$ | ||

${e}_{i}{e}_{j}$ | $=-{e}_{j}{e}_{i}\mathit{\hspace{1em}}(i\ne j)$ |

Trivially, $\mathrm{Cliff}({\mathbb{R}}^{0})=\mathbb{R}$, and it can be seen from the relations above that $\mathrm{Cliff}(\mathbb{R})\cong \u2102$, the complex numbers, and $\mathrm{Cliff}({\mathbb{R}}^{2})\cong \mathbb{H}$, the quaternions.

On the other ha nd, for $V={\u2102}^{n}$ we get the particularly answer of

$$\mathrm{Cliff}({\u2102}^{2k})\cong {\mathrm{M}}_{{2}^{k}}(\u2102)\mathit{\hspace{1em}\hspace{1em}}\mathrm{Cliff}({\u2102}^{2k+1})={\mathrm{M}}_{{2}^{k}}(\u2102)\oplus {\mathbf{M}}_{{2}^{k}}(\u2102).$$ |

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 |