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.
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
f : C(q) -> A such that the following diagram commutes
The Clifford algebra exists and can be constructed as follows: take the tensor algebra T(V) and mod out by the ideal generated by
Formal definition
V ----> C(q)
| /
| / Exists and is unique
| /
v v
A
i.e. such that fi = j.
It follows from this construction that i is injective, and V can be considered as a linear subspace of C(q).
Let
- B(u,v) = q(u+v) - q(u) - q(v)
- uv + vu = B(u,v)1
The Clifford algebra C(q) is filtered by subspaces
- dim C(q) = 2dim V.
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 2dim(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 ......
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 as follows...
The Clifford algebra is important in physics. Physicists usually consider the Clifford algebra to be spanned by matices γ1,...,γn which have the property that
- γi γj + γj γi = 2 ηi,j