K-monoid

Clifford algebra

Let (𝑉,𝑄) be a quadratic space over 𝕂. The Clifford algebra Cl⁡(𝑉,𝑄) is the freëst K-monoid generated by 𝑉 subject to the condition to geo

𝑣2=𝑄(𝑣)1

as formalized by the universal property. Away from 2, this is equivalent to the freëst unital associative algebra such that the anticommutator extends the polar form

{𝑣,𝑤}=𝑣𝑤+𝑤𝑣=𝑏𝑄(𝑣,𝑤)1

This motivates yet another perspective: Cl⁡(𝑉,𝑄) is the freëst unital associatve algebra whose associated Jordan algebra 𝐴+ has a product extending 𝑏𝑞, i.e. 𝐴+1/2 has a product extending ( −) ⋅( −).

In a sense the Clifford algebra generalizes, or rather quantizes the Exterior algebra. It is sometimes called the orthogonal Clifford algebra, as opposed to the related Weyl algebra which is sometimes called the symplectic Clifford algebra.

Universal property

Let (𝑉,𝑄) be a quadratic space over 𝕂. The associated Clifford algebra is a pair consisting of a K-monoid Cl⁡(𝑉,𝑄) and a linear map 𝜄 :𝑉 →Cl⁡(𝑉,𝑄) with the identity 𝜄(𝑣)2 =𝑄(𝑣)1 such that given any unital associative algebra 𝐴, a linear map 𝑓 :𝑉 →𝐴 satisfying 𝑓(𝑣)2 =𝑄(𝑣)1 factorizes uniquely through 𝜄

such that ¯𝑓 :Cl⁡(𝑉,𝑄) →𝐴 is a unital algebra homomorphism. This admits a unique extension to a functor Cl :𝖰𝖵𝖾𝖼𝗍𝕂 →𝖴𝖠𝗌𝖠𝗅𝗀𝕂 such that 𝜄 :1 ⇒Cl :𝖰𝖵𝖾𝖼𝗍𝕂 →𝖵𝖾𝖼𝗍𝕂 becomes a natural transformation.

Construction

The Clifford algebra may be constructed as a quotient algebra of the tensor algebra

Cl⁡(𝑉,𝑄)=𝑇∙𝑉⟨𝑣⊗𝑣−𝑄(𝑣):𝑣∈𝑉⟩⊴𝑇∙𝑉

where the divisor is the algebra ideal generated by tensors of the form 𝑣 ⊗𝑣 −𝑄(𝑣)1.

Proof of the universal property

proof

Relation to the exterior algebra

The exterior algebra is the associated graded algebra of the Clifford algebra, whence there is a natural linear isomorphism between them. With this identification, we have

𝑘⋀𝑗=1𝑣𝑗=1𝑘!∑𝜎∈𝑆𝑘sgn⁡(𝜎)𝑘∏𝑗=1𝑣𝑗

We carry over all the terminology, referring to 𝑘-vectors, &c.

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