Exterior Product

Prerequisites: Linear Algebra.

• Intuitive
• Concrete
• Abstract

Intuitive

The exterior product of two vectors can be pictured as the (oriented) area of the parallelogram enclosed by the two vectors. Similarly, the exterior product of three vectors can be pictured as the (oriented) volume of the parallelepiped enclosed by the three vectors.

The exterior product generalizes the cross product and (scalar) triple product from 3-dimensional space to spaces with any number of dimensions.

The exterior product is also known as the Grassmann product or wedge product.

Concrete

The exterior product is obtained by first taking the tensor product and then antisymmetrizing the result. The picture below shows how to do that for two 3D vectors (top), three 3D vectors (center), and a 3D vector and an antisymmetric tensor (bottom).

The exterior product of two 3D vectors is a rank-2 tensor with three independent components (shown in red), which match those of the cross product. The exterior product of three 3D vectors is a rank-3 tensor with only one independent component (shown in red), which matches the (scalar) triple product. The exterior product of a 3D vector and an antisymmetric tensor is again a rank-3 tensor with only one independent component (shown in red), which is related to the scalar product.

For a more detailed explanation of this picture see Fun with Symmetry.

Abstract

Let $V$ be a vector space of dimension $n$ over $\mathbb{R}$ (or any field of characteristic $0$). The exterior algebra is the quotient of the tensor algebra by the two-sided ideal generated by all $v\otimes v$, $$\Lambda(V)=T(V)/\langle v\otimes v : v\in V\rangle=\bigoplus_{k=0}^n\Lambda^kV,$$ and the exterior product $\wedge$ is the induced multiplication. It is associative, and since $v\wedge v=0$ for every $v\in V$, expanding $(v+w)\wedge(v+w)=0$ gives $v\wedge w=-w\wedge v$. For $\alpha\in\Lambda^kV$ and $\beta\in\Lambda^lV$ one has graded commutativity $\alpha\wedge\beta=(-1)^{kl}\beta\wedge\alpha$. If $e_1,\dots,e_n$ is a basis of $V$, the products $e_{i_1}\wedge\cdots\wedge e_{i_k}$ with $i_1<\cdots<i_k$ form a basis of $\Lambda^kV$, so $\dim\Lambda^kV=\binom{n}{k}$ and $\dim\Lambda(V)=2^n$.

Universal property. $\Lambda^kV$ is characterized by the property that alternating $k$-linear maps $V\times\cdots\times V\to W$ correspond one-to-one to linear maps $\Lambda^kV\to W$, via $v_1\wedge\cdots\wedge v_k\mapsto f(v_1,\dots,v_k)$. In characteristic $0$ one can identify $\Lambda^kV$ with the totally antisymmetric tensors in $V^{\otimes k}$, which is the description used in the Concrete layer. The identification involves a choice of normalization, for example $v\wedge w=v\otimes w-w\otimes v$ or $v\wedge w=\tfrac12(v\otimes w-w\otimes v)$, and factors of $k!$ in formulas depend on it.

Determinant and volume. A linear map $f:V\to V$ induces $\Lambda^kf$ with $\Lambda^kf(v_1\wedge\cdots\wedge v_k)=fv_1\wedge\cdots\wedge fv_k$. Since $\Lambda^nV$ is one-dimensional, $\Lambda^nf$ is multiplication by a number, and this number is $\det f$. In particular $v_1\wedge\cdots\wedge v_n=\det(A)\,e_1\wedge\cdots\wedge e_n$ when $v_j=\sum_iA_{ij}e_i$, which is the precise statement that the wedge product measures oriented volume. Vectors $v_1,\dots,v_k$ are linearly independent exactly when $v_1\wedge\cdots\wedge v_k\neq0$, and a nonzero decomposable $k$-vector determines its $k$-dimensional span up to scale. This gives the Plücker embedding of the Grassmannian of $k$-planes into the projective space $\mathbb{P}(\Lambda^kV)$.

Further structure. With an inner product and an orientation on $V$ there is the Hodge dual $\star:\Lambda^kV\to\Lambda^{n-k}V$; in three dimensions $\star(v\wedge w)=v\times w$, which is how the cross product arises. Applied pointwise to the cotangent spaces of a manifold, $\Lambda^kT^*M$ gives the bundle whose sections are the differential forms, on which the exterior derivative acts. Replacing the relation $v\otimes v=0$ by $v\otimes v=q(v)\,1$ for a quadratic form $q$ gives the Clifford algebra, which is isomorphic to $\Lambda(V)$ as a vector space (in characteristic different from $2$) but has a different product. The exterior algebra $\Lambda(\mathbb{C}^N)$ with generators $\theta_1,\dots,\theta_N$, $\theta_i\theta_j=-\theta_j\theta_i$, is the algebra of Grassmann numbers used in the path integral for fermions.

Contributing authors:

Leads to: Hodge Dual.

Authors: Eduard Sackinger