Clifford algebra

Clifford algebras are associative algebras of importance in mathematics, in particular in the theories of quadratic forms and of orthogonal groups, and in physics. They are named for William Kingdon Clifford.

Formal definition

Let V be a vector space over a field k, and

q : V → k

a quadratic form on V. The Clifford Algebra C(q) is a unital associative algebra over k together with a linear map

i : V → C(q)

defined by the following universal property:

for every associative algebra A over k with a linear map

j : V → A

such that for every v in V we have

j(v)² = q(v)1

(where 1 denotes the multiplicative identity of A), there is a unique algebra homomorphism

<math> f: C(q) \to A <math>

such that the following diagram commutes:

<math>\begin{matrix} V & \to & C(q) \\ \downarrow & \swarrow & \\ A && \end{matrix} <math>

i.e. such that fi = j.

The Clifford algebra exists and can be constructed as follows: take the tensor algebra T(V), and construct its quotient ring by the ideal generated by the elements

<math>v \otimes v - q(v)1<math>.

It follows from this construction that i is injective, and V can be considered as a linear subspace of C(q).

Let

<math> B ( u , v ) = q ( u + v ) - q ( u ) - q ( v ) \,<math>

be the bilinear form associated to q. It is a consequence of the definition that the identity

<math> uv + vu = B ( u , v )1 \,<math>

holds in C(q) for every pair (u, v) of vectors in V. If the field is of characteristic unequal to 2 this expression can be used as an alternative defining property.

The Clifford algebra C(q) is filtered by subspaces

<math> k \subseteq k + V \subseteq k + V + V^2 \subseteq \ldots \,<math>

of elements that can be written as monomials of 0, 1, 2, .. vectors in V. The associated graded algebra is canonically isomorphic to the exterior algebra <math> \bigwedge V <math> of the vectorspace. This shows in particular that

<math> \operatorname{dim} C( q ) = 2^{\operatorname{dim} V} \,<math>.

A more mundane way to see this is by choosing an arbitrary basis e_1, e_2, ..... for V. Using the anticommutation relation we can always express an element of the Clifford algebra as a linear combination of monomials of type

<math>e_{i_1} e_{i_2} e_{i_3} \cdots e_{i_n}, i_1 < i_2 <\cdots < i_n<math>

which gives an explicit isomorphism with the exterior algebra. Note that this is an isomorphism of vector spaces, not of algebras.

If V has finite even dimension, the field is algebraically closed and the quadratic form is non degenerate, the Clifford algebra is central simple. Thus by the Artin-Wedderburn theorem it is (non canonically) isomorphic to a matrix algebra. It follows that in this case C(q) has an irreducible representation of dimension 2^dim(V)/2 which is unique up to nonunique isomorphism. This is the (in)famous spinor representation, and its vectors are called spinors.

If dim V is odd ......               FILL IN!

In case the field k is the field of real numbers the Clifford algebra of a quadratic form of signature p,q is usually denoted C(p,q). These real Clifford algebras have been classified here.

The Clifford algebra is important in physics. Physicists usually consider the Clifford algebra to be spanned by matrices γ₁,...,γ_n which have the property that

<math> \gamma_i\gamma_j + \gamma_j\gamma_i = 2\eta_{i,j} \,<math>

where η is the matrix of a quadratic form of type p,q with respect to an orthonormal basis e₁,..., e_n. The γ_i matrix is nothing but the matrix of the multiplication by the vector e_i on the spinor representation with respect to some arbitrary basis of the spinors.