Editing Clifford algebra (section)

== Universal property and construction ==
Let {{math|''V''}} be a [[vector space]] over a [[field (mathematics)|field]]&nbsp;{{math|''K''}}, and let {{math|''Q'' : ''V'' → ''K''}} be a [[quadratic form]] on {{math|''V''}}.  In most cases of interest the field {{math|''K''}} is either the field of [[real number]]s&nbsp;{{math|'''R'''}}, or the field of [[complex number]]s&nbsp;{{math|'''C'''}}, or a [[finite field]].

A Clifford algebra {{math|Cl(''V'', ''Q'')}} is a pair {{math|(''B'', ''i'')}},{{efn|{{harvnb|Vaz|da Rocha|2016}} make it clear that the map {{math|''i''}} ({{math|''γ''}} in the quote here) is included in the structure of a Clifford algebra by defining it as "The pair {{math|(''A'', ''γ'')}} is a Clifford algebra for the quadratic space {{math|(''V'', ''g'')}} when {{math|''A''}} is generated as an algebra by {{math|{{mset| ''γ''('''v''') | '''v''' ∈ ''V''&nbsp;}}}} and {{math|{{mset| ''a''1{{sub|''A''}} | ''a'' ∈ '''R'''&nbsp;}}}}, and {{math|''γ''}} satisfies {{math|1=''γ''('''v''')''γ''('''u''') + ''γ''('''u''')''γ''('''v''') = 2''g''('''v''', '''u''')}} for all {{math|'''v''', '''u''' ∈ ''V''}}."}}{{sfn|Lounesto|1996|pages=3–30|ps=&nbsp;or [https://users.aalto.fi/~ppuska/mirror/Lounesto/counterexamples.htm abridged version]}} where {{math|''B''}} is a [[unital algebra|unital]] [[associative algebra]] over {{math|''K''}} and {{math|''i''}} is a [[linear transformation|linear map]] {{math|''i'' : ''V'' → ''B''}} that satisfies {{math|1=''i''(''v'')<sup>2</sup> = ''Q''(''v'')1{{sub|''B''}}}} for all {{math|''v''}} in {{math|''V''}}, defined by the following [[universal property]]: given any unital associative algebra {{math|''A''}} over {{math|''K''}} and any linear map {{math|''j'' : ''V'' → ''A''}} such that
<math display="block">j(v)^2 = Q(v)1_A \text{ for all } v \in V</math>
(where {{math|1<sub>''A''</sub>}} denotes the multiplicative identity of {{math|''A''}}), there is a unique [[algebra homomorphism]] {{math|''f'' : ''B'' → ''A''}} such that the following diagram [[commutative diagram|commutes]] (i.e. such that {{math|1=''f'' ∘ ''i'' = ''j''}}):

<div style="text-align: center;">[[Image:CliffordAlgebra-01.png]]</div>

The quadratic form {{math|''Q''}} may be replaced by a (not necessarily symmetric{{sfn|Lounesto|1993|ps=none}}) [[bilinear form]] {{math|{{angle brackets|⋅,⋅}}}} that has the property {{math|1={{angle brackets|''v'', ''v''}} = ''Q''(''v''), ''v'' ∈ ''V''}}, in which case an equivalent requirement on {{math|''j''}} is
<math display="block"> j(v)j(v) = \langle v, v \rangle 1_A \quad \text{ for all } v \in V .</math>

When the characteristic of the field is not {{math|2}}, this may be replaced by what is then an equivalent requirement,
<math display="block"> j(v)j(w) + j(w)j(v) = ( \langle v, w \rangle + \langle w, v \rangle )1_A \quad \text{ for all } v, w \in V , </math>
where the bilinear form may additionally be restricted to being symmetric without loss of generality.

A Clifford algebra as described above always exists and can be constructed as follows: start with the most general algebra that contains {{math|''V''}}, namely the [[tensor algebra]] {{math|''T''(''V'')}}, and then enforce the fundamental identity by taking a suitable [[quotient ring|quotient]]. In our case we want to take the two-sided [[Ideal (ring theory)|ideal]] {{math|''I<sub>Q</sub>''}} in {{math|''T''(''V'')}} generated by all elements of the form
<math display="block">v\otimes v - Q(v)1</math> for all <math>v\in V</math>
and define {{math|Cl(''V'', ''Q'')}} as the quotient algebra
<math display="block">\operatorname{Cl}(V, Q) = T(V) / I_Q .</math>

The [[Ring (mathematics)|ring]] product inherited by this quotient is sometimes referred to as the '''Clifford product'''{{sfn|Lounesto|2001|loc=§1.8|ps=none}} to distinguish it from the exterior product and the scalar product.

It is then straightforward to show that {{math|Cl(''V'', ''Q'')}} contains {{math|''V''}} and satisfies the above universal property, so that {{math|Cl}} is unique up to a unique isomorphism; thus one speaks of "the" Clifford algebra {{math|Cl(''V'', ''Q'')}}. It also follows from this construction that {{math|''i''}} is [[injective function|injective]]. One usually drops the&nbsp;{{math|''i''}} and considers {{math|''V''}} as a [[linear subspace]] of {{math|Cl(''V'', ''Q'')}}.

The universal characterization of the Clifford algebra shows that the construction of {{math|Cl(''V'', ''Q'')}} is {{em|functorial}} in nature. Namely, {{math|Cl}} can be considered as a [[functor]] from the [[category (mathematics)|category]] of vector spaces with quadratic forms (whose [[morphism]]s are linear maps that preserve the quadratic form) to the category of associative algebras. The universal property guarantees that linear maps between vector spaces (that preserve the quadratic form) extend uniquely to algebra homomorphisms between the associated Clifford algebras.