[[K-monoid]]
# Exterior algebra

The **exterior algebra** $\bigwedge^• V$ of a [[vector space]] $V$ is the freëst [[Alternating multilinear map|alternating]] [[K-monoid]] containing $V$, #m/def/falg 
as formalized by the [[#Universal property]].

The exterior algebra is in a sense generalized by, or rather _quantized_ by, the [[Clifford algebra]].
Conceptually similar is the [[Symmetric algebra]].

## Universal property

Let $V$ be a [[vector space]] over $\mathbb{K}$ the associated **exterior algebra** is a pair consisting of an [[Alternating multilinear map|alternating]] [[K-monoid]] ${\textstyle\bigwedge}^\bullet V$ and a [[linear map]] $\iota : V \to {\textstyle\bigwedge}^\bullet V$ 
such that given any unital associative algebra $A$,
a linear map $f : V \to A$ satisfying the identity $f(v)^2 = 0$ factorizes uniqely through $\iota$

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such that $\bar{f} : {\textstyle\bigwedge}^\bullet V \to A$ is a [[Algebra homomorphism|unital algebra homomorphism]].
This admits a unique extension to a [[functor]] ${\textstyle\bigwedge}^\bullet : \Vect_{\mathbb{K}} \to \cat{UAsAlg}_{\mathbb{K}}$ such that $\iota: 1 \Rightarrow {\textstyle\bigwedge}^\bullet : \Vect_{\mathbb{K}} \to \Vect_{\mathbb{K}}$ becomes a [[natural transformation]].

## Construction

The exterior  algebra may be constructed as a [[Quotient algebra|quotient]] of the [[tensor algebra]]
$$
\begin{align*}
{\textstyle\bigwedge}^• V = \frac{T^•V}{\langle v \otimes v : v \in V \rangle_{\trianglelefteq T^\bullet V} }
\end{align*}
$$
where the divisor is the [[algebra ideal]] generated by tensors of the form $v \otimes v$,
where the **wedge product** $v \wedge w$ is the [[quotient algebra]] product.

> [!missing]- Proof of universal property
> #missing/proof

## Graded structure
Like the tensor algebra, the exterior algebra is $\mathbb{N}_{0}$-[[Graded algebra|graded]] into **exterior powers**
$$
\begin{align*}
{\textstyle\bigwedge}^•V = {\textstyle\bigwedge}^0 V \oplus {\textstyle\bigwedge}^1 V \oplus {\textstyle\bigwedge}^2 V \oplus \dots
\end{align*}
$$
such that $\bigwedge^k V \wedge \bigwedge^p V \sube \bigwedge^{k+p} V$.
If $\{ e_{i} \}_{i=1}^n$ is a basis for $V$, then 
$$
\{ e_{i_{1}} \wedge e_{i_{2}} \wedge \dots \wedge e_{i_{k}} \mid 1 \leq i_{1} < i_{2} < \dots < i_{k} \leq n \}
$$
is a basis for $\bigwedge^kV$, hence
$$
\begin{align*}
\dim {\textstyle\bigwedge}^k V = {n \choose k}
\end{align*}
$$
Elements of the form $\bigwedge^k_{i=1}v_{k}$ where $v_{k} \in V$ are called **$k$-blades**,
whereas **$k$-vectors** are in general linear combinations of $k$-blades.
The distinction is the same as that of separable and entangled tensors.
In particular, if $\dim V = n$ then

- An $(n-1)$-vector is a [[pseudovector]] ($\dim \bigwedge^{n-1} V = n$)
- An $n$-vector is a [[pseudoscalar]] ($\dim \bigwedge^n V = 1$)

## Geometric interpretation

Geometrically, the magnitude of a $k$-blade represents the $k$-hypervolume of the $k$-hyperparallelotope spanned defined by some vectors.
Hence it generalizes the [[cross product]],
which can be thought of as resulting from the linear isomorphism from ${\textstyle\bigwedge}^k \mathbb{K}^3$ to $\mathbb{K}^3$,
which is natural if $\mathbb{K}^3$ is taken as an [[oriented vector space]].

## Relation to antisymmetric tensors

Let $\pi : T^\bullet V \to {\textstyle\bigwedge}^\bullet V$ be the [[Homomorphism of graded vector spaces#^graded]] natural projection.
If $k!$ is invertible in the ground field, in particular if [[Characteristic|$\opn{char}\mathbb K = 0$]], then
${\textstyle\bigwedge}^k V$ may be identified as a vector space with the subspace $T_{-}^k V$ of [[Tensor algebra|$T^k V$]] consisting of [[Antisymmetric tensor|antisymmetric tensors]], where we identify
$$
\begin{align*}
\bigwedge_{i=1}^k v_{k} = \frac{1}{k!} \sum_{\sigma \in S_{k}} \sgn(\sigma) \bigotimes_{j=1}^k v_{\sigma(j)}
\end{align*}
$$
or more generally letting $\tilde{\pi} = \pi \restriction T^\bullet_{-}V$,
for homogenous vectors $v,w \in {\textstyle\bigwedge}^\bullet V$
$$
\begin{align*}
\tilde{\pi}^{-1} (v \wedge w) = \frac{v \otimes w - w \otimes v}{\deg v + \deg w} = \frac{v \otimes_{-} w}{\deg v + \deg w}
\end{align*}
$$

## Properties

- The $k$-blades satisfy the [[Plücker relations]], enabling the [[Plücker embedding]] of the [[Grassmannian]] $\mathrm{Gr}_{k}(V)$ into the [[Projectivization]] $\mathrm{P}(\bigwedge^k V)$

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