Editing Clifford algebra (section)

== Introduction and basic properties ==
A Clifford algebra is a [[unital algebra|unital]] [[associative algebra]] that contains and is generated by a [[vector space]] {{math|''V''}} over a [[Field (mathematics)|field]] {{math|''K''}}, where {{math|''V''}} is equipped with a [[quadratic form]] {{math|''Q'' : ''V'' → ''K''}}. The Clifford algebra {{math|Cl(''V'', ''Q'')}} is the "freest" unital associative algebra generated by {{math|''V''}} subject to the condition{{efn|Mathematicians who work with real Clifford algebras and prefer positive definite quadratic forms (especially those working in [[index theory]]) sometimes use a different [[sign convention|choice of sign]] in the fundamental Clifford identity. That is, they take {{math|1=''v''<sup>2</sup> = −''Q''(''v'')}}. One must replace {{math|''Q''}} with {{math|−''Q''}} in going from one convention to the other.}}
<math display="block">v^2 = Q(v)1\ \text{ for all } v\in V,</math>
where the product on the left is that of the algebra, and the {{math|1}} on the right is the algebra's [[multiplicative identity]] (not to be confused with the multiplicative identity of {{math|''K''}}).  The idea of being the "freest" or "most general" algebra subject to this identity can be formally expressed through the notion of a [[universal property]], as done [[#Universal property and construction|below]].

When {{math|''V''}} is a finite-dimensional real vector space and {{math|''Q''}} is [[nondegenerate quadratic form|nondegenerate]], {{math|Cl(''V'', ''Q'')}} may be identified by the label {{math|Cl<sub>''p'',''q''</sub>('''R''')}}, indicating that {{math|''V''}} has an orthogonal basis with {{math|''p''}} elements with {{math|1=''e''<sub>''i''</sub><sup>2</sup> = +1}}, {{math|''q''}} with {{math|1=''e<sub>i</sub>''<sup>2</sup> = −1}}, and where {{math|'''R'''}} indicates that this is a Clifford algebra over the reals; i.e. coefficients of elements of the algebra are real numbers. This basis may be found by [[orthogonal diagonalization]].

The [[free algebra]] generated by {{math|''V''}} may be written as the [[tensor algebra]] {{math|⨁<sub>''n''≥0</sub> ''V'' ⊗ ⋯ ⊗ ''V''}}, that is, the [[direct sum]] of the [[tensor product]] of {{math|''n''}} copies of {{math|''V''}} over all {{math|''n''}}. Therefore one obtains a Clifford algebra as the [[Quotient ring|quotient]] of this tensor algebra by the two-sided [[Ideal (ring theory)|ideal]] generated by elements of the form {{math|''v'' ⊗ ''v'' − ''Q''(''v'')1}} for all elements {{math|''v'' ∈ ''V''}}. The product induced by the tensor product in the quotient algebra is written using juxtaposition (e.g. {{math|''uv''}}).  Its associativity follows from the associativity of the tensor product.

The Clifford algebra has a distinguished [[Linear subspace|subspace]]&nbsp;{{math|''V''}}, being the [[Image (mathematics)|image]] of the [[embedding]] map. Such a subspace cannot in general be uniquely determined given only a {{math|''K''}}-algebra that is [[isomorphic]] to the Clifford algebra.

If {{math|2}} is [[Unit (ring theory)|invertible]] in the ground field {{math|''K''}}, then one can rewrite the fundamental identity above in the form
<math display="block">uv + vu = 2\langle u, v\rangle1\ \text{ for all } u,v \in V,</math>
where
<math display="block">\langle u, v \rangle = \frac{1}{2} \left( Q(u + v) - Q(u) - Q(v) \right)</math>
is the [[symmetric bilinear form]] associated with {{math|''Q''}}, via the [[polarization identity]].

Quadratic forms and Clifford algebras in characteristic {{math|2}} form an exceptional case in this respect. In particular, if {{math|1=char(''K'') = 2}} it is not true that a quadratic form necessarily or uniquely determines a symmetric bilinear form that satisfies {{math|1=''Q''(''v'') = {{angle brackets|''v'', ''v''}}}},{{sfn|Lounesto|1993|pp=155–156|ps=none}} Many of the statements in this article include the condition that the characteristic is not {{math|2}}, and are false if this condition is removed.

=== As a quantization of the exterior algebra ===
Clifford algebras are closely related to [[exterior algebra]]s. Indeed, if {{math|1=''Q'' = 0}} then the Clifford algebra {{math|Cl(''V'', ''Q'')}} is just the exterior algebra {{math|⋀''V''}}. Whenever {{math|2}} is invertible in the ground field&nbsp;{{math|''K''}}, there exists a canonical ''linear'' isomorphism between {{math|⋀''V''}} and {{math|Cl(''V'', ''Q'')}}. That is, they are [[naturally isomorphic]] as vector spaces, but with different multiplications (in the case of characteristic two, they are still isomorphic as vector spaces, just not naturally). Clifford multiplication together with the distinguished subspace is strictly richer than the [[exterior product]] since it makes use of the extra information provided by&nbsp;{{math|''Q''}}.

The Clifford algebra is a [[filtered algebra]]; the [[associated graded algebra]] is the exterior algebra.

More precisely, Clifford algebras may be thought of as ''quantizations'' (cf. [[quantum group]]) of the exterior algebra, in the same way that the [[Weyl algebra]] is a quantization of the [[symmetric algebra]].

Weyl algebras and Clifford algebras admit a further structure of a [[*-algebra]], and can be unified as even and odd terms of a [[superalgebra]], as discussed in [[CCR and CAR algebras]].