[[K-monoid]]
# Clifford algebra

Let $(V, Q)$ be a [[quadratic space]] over $\mathbb{K}$. 
The **Clifford algebra** $\opn{Cl}(V,Q)$ is the freëst [[K-monoid]] generated by $V$ subject to the condition to #m/def/falg/geo
$$
\begin{align*}
v^2 = Q(v)1
\end{align*}
$$
as formalized by the [[#universal property]].
[[Away from 2]], this is equivalent to the freëst unital associative algebra such that the [[anticommutator]] extends the [[Quadratic form|polar form]]
$$
\begin{align*}
\{ v,w \} = vw + wv = b_{Q}(v,w)1
\end{align*}
$$
This motivates yet another perspective: $\Cl(V,Q)$ is the freëst unital associatve algebra whose [[Anticommutator|associated Jordan algebra]] $A^+$ has a product extending $b_{q}$,
i.e. $A^{+ 1/2}$ has a product extending $(-)\cdot (-)$.

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 $\mathbb{K}$.
The associated **Clifford algebra** is a pair consisting of a [[K-monoid]] $\Cl(V,Q)$ and a [[linear map]] $\iota : V \to \Cl(V,Q)$ with the identity $\iota(v)^2 = Q(v)1$ 
such that given any unital associative algebra $A$, a linear map $f : V \to A$ satisfying $f(v)^2 = Q(v)1$ factorizes uniquely through $\iota$

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such that $\bar f: \Cl(V,Q) \to A$ is a [[Algebra homomorphism|unital algebra homomorphism]].
This admits a unique extension to a [[functor]] $\Cl : \cat{QVect}_{\mathbb{K}} \to \cat{UAsAlg}_{\mathbb{K}}$ such that $\iota : 1 \Rightarrow \Cl : \cat{QVect}_{\mathbb{K}} \to \Vect_{\mathbb{K}}$ becomes a [[natural transformation]].

## Construction

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

$$
\begin{align*}
\opn{Cl}(V,Q) = \frac{T^\bullet V}{\langle v \otimes v - Q(v) : v \in V \rangle_{\trianglelefteq T^\bullet V} }
\end{align*}
$$
where the divisor is the [[algebra ideal]] generated by tensors of the form $v \otimes v - Q(v)1$.

> [!missing]- Proof of the universal property
> #missing/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 filtered algebra and associated graded algebra|natural linear isomorphism]] between them.
With this identification, we have
$$
\begin{align*}
\bigwedge_{j=1}^k v_{j} = \frac{1}{k!} \sum_{\sigma \in S_{k}} \sgn(\sigma) \prod_{j=1}^k v_{j}
\end{align*}
$$
We carry over all the terminology, referring to $k$-vectors, &c.

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