Clifford algebra

Let V be a vector spaceMathworldPlanetmath over a field k, and Q:V×Vk a symmetric bilinear formMathworldPlanetmath. Then the Clifford algebraMathworldPlanetmathPlanetmath Cliff(Q,V) is the quotient of the tensor algebra 𝒯(V) by the relations

vw+wv=-2Q(v,w)  v,wV.

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

In addition, we do have a filtration on Cliff(Q,V) (making it a filtered algebra), and the associated graded algebraMathworldPlanetmath GrCliff(Q,V) is simply Λ*V, the exterior algebraMathworldPlanetmath of V. In particular,


The most commonly used Clifford algebra is the case V=n, and Q is the standard inner productMathworldPlanetmath with orthonormal basisMathworldPlanetmath e1,,en. In this case, the algebraPlanetmathPlanetmath is generated by e1,,en and the identity of the algebra 1, with the relations

ei2 =-1
eiej =-ejei(ij)

Trivially, Cliff(0)=, and it can be seen from the relations above that Cliff(), the complex numbers, and Cliff(2), the quaternions.

On the other ha nd, for V=n we get the particularly answer of

Cliff(2k)M2k()  Cliff(2k+1)=M2k()𝐌2k().
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