oSubjects: Linear Algebra, Differential Geometry, Clifford Algebra
Links: Tensor Product of Modules, Algebra Ideals and Quotient Algebras, Graded Ring, Dual Vector Spaces, Exterior Algebra of Multicovectors, Symmetric Group, Dual Vector Spaces

Def: The exterior algebra of a vector space $V$ over a field $K$ is defined as the quotient algebra of the tensor algebra $T(V)$, where $$T(V) = \bigoplus_{k = 0}^\infty T^k V$$by the two-sided ideal $I$ generated by all the elements of the form $x\otimes x$ such that $x\in V$. Symbolically, $$\bigwedge (V) := T(V)/I$$
The product defined on $\bigwedge (V)$ is called the exterior product and it is defined by $$\alpha \wedge \beta = \alpha \otimes \beta + I$$
Prop: The exterior product is anticommutative on elements of $V$. If $x,y\in V$, then $x\wedge y=-y\wedge x$.

Prop: Let $x_1,\dots ,x_k\in V$, and $\sigma \in S_k$ then $$x_{\sigma(1)}\wedge \dots \wedge x_{\sigma(n)} = \text{sgn}(\sigma) x_1 \wedge \dots \wedge x_k.$$

Exterior Powers

Def: If $\alpha \in \bigwedge (V)$ is of the form $\alpha =v_1\wedge v_2\wedge \cdots \wedge v_k$, where $v_1,v_2,\dots ,v_k\in V$, then $\alpha$ said to be decomposable, simple, or a blade. With this the $k$th exterior power of $V$, denoted as $\underset{k}{\bigwedge }(V)$, is the vector subspace $\bigwedge (V)$ defined as: $${\textstyle\bigwedge}{k}(V):= \text{span}{v_1\wedge v_2 \wedge \dots \wedge v_k \mid v_i \in V, i \in {1, \dots, k}}$$
Prop: If $V$ is a finite dimensional vector space of dimension $n$ and $\{e_1,\dots ,e_n\}$ is a basis for $V$, then the set $${e\wedge e_{i_2}\wedge \dots \wedge e_{i_k} \mid 1 \le i_1 < i_2 < \dots < i_k \le n}$$is a basis for ${\bigwedge }_k(V)$.

Cor: The dimension of ${\bigwedge }_k(V)$ can be calculated and $$\dim {\textstyle\bigwedge}_{k}(V) = {n \choose k},$$where $n$ is the dimension of $V$, and $k$ is the number of vectors in the product.

Cor: If $k>n$, then ${\bigwedge }_k(V)=\{0\}$.

Prop: Any element of the exterior algebra can be written as the sum of $k$-vectors. Hence, as a vector space the exterior algebra is the direct sum $${\textstyle\bigwedge}(V) = \bigoplus_{k = 0}^n {\textstyle\bigwedge}_{k}(V)$$where by convention, ${\bigwedge }_0(V)=K$, the field underlying $V$ and ${\bigwedge }_1(V)=V$, and its dimension is equal to $2^n$.

Prop: $\bigwedge (V)$ is a graded algebra, meaning if $\alpha \in {\bigwedge }^k(V)$ and $\beta \in {\bigwedge }^{\ell }(V)$, then $\alpha \wedge \beta \in {\bigwedge }^{k+\ell }(V)$, and $$\alpha \wedge \beta = (-1)^{k\ell} \beta\wedge \alpha.$$

Universal Properties

Universal property of the exterior algebra: Given any unital $K$-algebra $A$ and any $K$-linear map $f:V\to A$ such that $f(v)f(v)=0$ for every $v\in V$, then there exists a precisely one unital algebra homomorphism $\phi :\bigwedge (V)\to A$ such that $f(v)=\phi (i(v))$ for all $v\in V$, and $i$ is the natural inclusion of $V$ in $\bigwedge (V)$.

\usepackage{tikz-cd} 
\begin{document} 
\begin{tikzcd}[row sep=2cm, column sep=2cm]
V \arrow[r,hook,"i"] \arrow[dr, "f"'] & {\textstyle\bigwedge}(V) \arrow[d, dashed,"\varphi"]\\ 
& A
\end{tikzcd}
\end{document}

Obs: There's a natural map $w:V^k\to {\bigwedge }_k(V)$, defined by $w(v_1,\dots ,v_k)=v_1\wedge \cdots \wedge v_k$, this map is an alternating $k$-linear map.

Universal property of the exterior powers: Given two vector spaces $V$ and $W$ and a natural number $k$, an alternating $k$-linear map from is a function from $V^k$ to $W$, $f:V^k\to W$, with the property that $(\sigma f)=\text{sgn}(\sigma )f$. Then there exists a unique linear map $\varphi :{\bigwedge }_k(V)\to X$ with $f=\varphi \circ w$.

\usepackage{tikz-cd} 
\begin{document} 
\begin{tikzcd}[row sep=2cm, column sep=2cm]
V^k \arrow[r,hook,"w"] \arrow[dr, "f"'] & {\textstyle\bigwedge}_k(V) \arrow[d, dashed,"\varphi"]\\ 
& W
\end{tikzcd}
\end{document}

This universal property characterise the space of alternating $k$-linear maps on $V$ and serve as a definition.

Duality

Given any $K$-vector space $V$, and $k$ linear forms $f_1,\dots ,f_k\in V^{\ast }$ and $k$ elements $v_1,\dots ,v_k\in V$, we first form a $k\times k$ matrix $A$ with entries $A_{ij}:=f_i(v_j)$. Now any determinant is an alternating multilinear function of its rows and and of its columns. Hence, for a fixed list $f_1,\dots ,f_k$ of linear form $detA$ is an alternating $k$-linear function of $v_1,\dots ,v_k$. With this in mind, we can define a function $\mathrm{\Psi }:{\bigwedge }_k(V^{\ast })\to {\bigwedge }^k(V)=({\bigwedge }_k(V){)}^{\ast },$ as $$\Psi(f_1\wedge\dots\wedge f_k)(v_1,\dots, v_k) := \det(f_i(v_j)). $$

Th: If $V$ is a finite dimensional $K$-vector space, then for each natural number $k$ the linear transformation $\mathrm{\Psi }:{\bigwedge }_k(V^{\ast })\to {\bigwedge }^k(V)=({\bigwedge }_k(V){)}^{\ast }$ is a vector space isomorphism.

Th: If $V$ is an $n$-dimensional $K$-vector space with a chosen isomorphism $\eta :{\bigwedge }_n(V)\to K$, then the bilinear map ${\bigwedge }_{n-k}(V)\times {\bigwedge }_k(V)\to K$ given for $s\in {\bigwedge }_{n-k}(V)$ and $t\in {\bigwedge }_k(V)$, by $(s,t)\mapsto \eta (s\wedge t)$ induces an isomorphism $$\theta: {\textstyle \bigwedge}_{n-k}(V) \to {\textstyle \bigwedge}^{! k}(V),\qquad (\theta s)t = \eta(s \wedge t) $$This theorem is telling us that if we have an orientation we can have an isomorphism between ${\bigwedge }_{n-k}(V)$ and ${\bigwedge }^k(V)$.

Th: If $U$ is a finite-dimensional inner product space over the field $K$, then for each $k$, ${\bigwedge }_k(U)$ is an inner product space with an inner product given, for two list $v$ and $w$ of $k$ vectors as $$\langle v_1\wedge \dots \wedge v_k, w_1\wedge\dots \wedge w_k\rangle := \det(\langle v_i, w_j\rangle), $$where we remark that we are calculating the determinant to something similar to a Gram matrix.

Prop: If $V$ is a finite dimensional inner product space of dimension $n$ and $\{e_1,\dots ,e_n\}$ is an orthonormal basis for $V$, then the set $${e_{i_1}\wedge e_{i_2}\wedge \dots \wedge e_{i_k} \mid 1 \le i_1 < i_2 < \dots < i_k \le n}$$is an orthonormal basis for ${\bigwedge }_k(V)$.