Wedge product/Universal property/Section

The following statement gives the universal property of the wedge product.

Theorem

Let $K$ be a field, let $V$ be a $K$-vector space, and $n\in \mathbb{N}$. Let

$\psi :V^n⟶W$

be an alternating multilinear mapping in another $K$-vector space $W$. Then there exists a uniquely determined linear mapping

$\tilde{\psi }:\stackrel{n}{\bigwedge }V⟶W$

such that the diagram

$\begin{array}{cccc}{V}^n& \stackrel{}{⟶}& \stackrel{n}{\bigwedge }V& \\ & \searrow & \downarrow & \\ & & W& \end{array}$

commutes.

Proof

We use the notation from fact. The assignment

$e_{(v_1,\dots ,v_n)}⟼\psi (v_1,\dots ,v_n)$

defines, due to fact, a $K$-linear mapping

$\overline{\psi }:H⟶W.$

As $\psi$ is multilinear and alternating, $\overline{\psi }$ sends the linear subspace $U\subseteq H$ to $0$. According to fact, there exists a $K$-linear mapping

$\tilde{\psi }:H/U⟶W,$

which is compatible with $\overline{\psi }$.
The uniqueness follows from the fact that the $v_1\wedge \dots \wedge v_n$ form a generating system of $\stackrel{n}{\bigwedge }V$, and these have to be sent to $\psi (v_1,\dots ,v_n)$.

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We denote by ${\mathrm{Alt}}^n(V,W)$ the set of all alternating mappings from $V^n$ to $W$. This set carries a natural $K$-vector space structure; see exercise.

Corollary

Let $K$ be a field, let $V$ and $W$ be $K$-vector spaces, and $n\in \mathbb{N}$. Then there exists a natural isomorphism

${\mathrm{Hom}}_K(\stackrel{n}{\bigwedge }V,W)⟶{\mathrm{Alt}}^n(V,W),\psi ⟼((v_1,\dots ,v_n)\mapsto \psi (v_1\wedge \dots \wedge v_n)).$

Proof

The mapping is just the composition $\psi \mapsto \psi \circ \delta$, where $\delta :V\times \cdots \times V\to \stackrel{n}{\bigwedge }V$ denotes the canonical mapping. The linearity of the assignment comes from the linear structures on the space of homomorphisms, and on the space of alternating mappings. The bijectivity of the mapping follows from fact.

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Corollary

Let $K$ be a field, let $V$ be a $K$-vector space, and $n\in \mathbb{N}$. Then there exists a natural isomorphism

${\left(\stackrel{n}{\bigwedge }V\right)}^{\ast }⟶{\mathrm{Alt}}^n(V,K),\psi ⟼((v_1,\dots ,v_n)\mapsto \psi (v_1\wedge \dots \wedge v_n)).$

Proof

This is a special case of fact.

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Theorem

Let $K$ be a field, and let $V$ denote a $K$-vector space, and $k\in {\mathbb{N}}_{+}$. Then there exists a canonical surjective linear mapping

$V{\otimes }_{}\cdots {\otimes }_{}V⟶V\wedge \dots \wedge V,v_1{\otimes }_{}\cdots {\otimes }_{}{v}_k⟼v_1\wedge \dots \wedge v_k.$

Proof

The multilinear mapping

$V^k⟶\stackrel{k}{\bigwedge }V,(v_1,\dots ,v_k)⟼v_1\wedge \dots \wedge v_k,$

induces, due to fact  (2), the linear mapping. The surjectivity rests on the fact that the generating system $v_1\wedge \dots \wedge v_k$ belongs to the image.

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