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
V > C(q)  /  / Exists and is unique  / v v A
i.e. such that fi = j.
The Clifford algebra exists and can be constructed as follows: take the tensor algebra[?] T(V) and mod out by the ideal generated by
Let
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
k subset k + V subset k + V + V^{2} subset ...
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[?] Λ V of the vectorspace. This shows in particular that
e_{i_1} e_{i_2} e_{i_3} ... e_{i_n} i_1 < i_2 .... < i_n
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 ......
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 γ_{1},...,γ_{n} which have the property that
γ_{i} γ_{j} + γ_{j} γ_{i} = 2 η_{i,j}
where η is the matrix of a quadratic form of type p,q with respect to an orthonormal basis e_{1},..., 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.
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