PlanetMath: Clifford algebra

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Clifford algebra

(Definition)

Let be a vector space over a field , 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

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

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 . In particular,

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

The most commonly used Clifford algebra is the case $V=\mathbb{R}^n$ , and 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 , with the relations

$\displaystyle e_i^2$	$\displaystyle =-1$
$\displaystyle e_ie_j$	$\displaystyle =-e_je_i \quad (i\neq j)$

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 simple answer of

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

"Clifford algebra" is owned by rmilson. [ full author list (2) | owner history (1) ]

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Cross-references: quaternions, complex numbers, identity, generated by, algebra, orthonormal basis, inner product, exterior algebra, graded algebra, filtered algebra, filtration, addition, superalgebra, homogeneous, relations, tensor algebra, quotient, symmetric bilinear form, field, vector space
There are 5 references to this entry.

This is version 6 of Clifford algebra, born on 2002-12-21, modified 2005-10-19.
Object id is 3803, canonical name is CliffordAlgebra2.
Accessed 8249 times total.

Classification:

AMS MSC:	11E88 (Number theory :: Forms and linear algebraic groups :: Quadratic spaces; Clifford algebras)
	15A66 (Linear and multilinear algebra; matrix theory :: Clifford algebras, spinors)

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