word looked up :

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. The formal definition follows. 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

               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

v tensor v - q(v)² 1.

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)

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

uv + vu = B(u,v)1

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 + V² 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

dim C(q) = 2^{dim V}.

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

     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 γ₁,...,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 e₁,..., e_n. The γ_i matrix isn'thing but the matrix of the multiplication by the vector e_i on the spinor representation with respect to some arbitrary basis of the spinors.

from the public domain (does not contain a notice indicating that it is and distributed to anyone in the United States without paying any fees with the phrase "Project gutenberg.html">Gutenberg" associated with or appearing on the through 1.E.7 or obtain permission for the use of the work and the 1.E.9. 1.E.3. If an individual Project Gutenberg-tm/tm.html">tm/tm.html">tm/tm.html">tm/tm.html">tm/tm.html">tm/tm.html">tm/tm.html">tm electronic.html">electronic.html">electronic work is posted must comply with both paragraphs 1.E.1 through 1.E.7 and any additional to the Project Gutenberg-tm License for all works posted with the License terms.html">terms from this work, or any files containing a part of this electronic work, or any part of this electronic work, without active links or immediate access.html">access to the full terms of the Project compressed, marked up, nonproprietary or proprietary form, including any distribute copies of a Project Gutenberg-tm work in a format.html">format other than posted on the official Project Gutenberg-tm web site (www.gutenberg.net), copy, a means of exporting a copy, or a means of obtaining a copy upon form. Any alternate format must include the full Project Gutenberg-tm performing, copying or distributing any Project Gutenberg-tm works access to or distributing Project Gutenberg-tm electronic works provided the use of Project Gutenberg-tm works calculated using the method owed to the owner of the Project Gutenberg-tm trademark, but he Project Gutenberg Literary Archive Foundation. Royalty payments prepare (or are legally required to prepare) your periodic tax sent to the Project Gutenberg Literary Archive Foundation at the .

On wordlookup.net

All is still licensed under the GNU FDL.
It uses material from the wikipedia.

navig stuff