n-dimensional "cross product" reference request

Question

I have written a paper which involves a "cross product" in $\mathbb{R}^n$ and I would like to have a reference to point to.

Let ${\bf e_1}, \dots, {\bf e_n}$ be the standard basis for $\mathbb{R}^n$ and let ${\bf w_1} = (w_{11},\dots,w_{1n}), \dots, {\bf w_{n-1}}=(w_{n-1\;1},\dots,w_{n-1\;n}) \in \mathbb{R}^n$. Then one can define

$${\bf w_1} \times {\bf w_2} \times \cdots \times {\bf w_{n-1}} = \begin{vmatrix} {\bf e_1} & {\bf e_2} & \cdots & {\bf e_n} \cr w_{11} & w_{12} & \cdots & w_{1n} \cr \vdots & \vdots & & \vdots \cr w_{n-1\;1} & w_{n-1\;2} & \cdots & w_{n-1\;n} \end{vmatrix}$$

where the right hand side is a "determinant".

Note: One can express this "cross product" in terms of exterior algebra operations. It is equivalent to $*({\bf w_1} \wedge {\bf w_2} \wedge \cdots \wedge {\bf w_{n-1}})$ where "$*$" is the Hodge dual operator.

Obviously if ${\bf a}, {\bf b} \in \mathbb{R}^3$, then ${\bf a} \times {\bf b}$ is the regular cross product and this $(n-1)$-airy product has the same properties as the regular cross product (it is a vector perpendicular to the vectors being multiplied and the length of this vector is given by the $(n-1)$-volume of the parallelotope spanned by the ${\bf w}$'s).

It was pointed out that this product appears in Susan Colley's "Vector Calculus" text [I have a second edition where this product is explored in problems 29-31 in section 1.6 on "$n$-dimensional geometry"]. Colley said she didn't have a good reference to point to (she couldn't recall where she'd seen it before).

I guess I could just refer to her book, but was wondering if anyone knows a better/historical reference? Or if not a paper does anyone know who I should attribute this to?

Alternatively, it would be nice to know if there is no "original" reference to point to and that this is just common knowledge/folklore.

Thank you!

Answer 1

Here's a bit of history on cross products that, if not directly useful, will hopefully provide some context. They were defined in "Beno Eckmann, Stetige Losungen linearer Gleichungssysteme, Comment. Math. Helv. 15(1943)" as follows: An $r$-fold cross product on a real vector space $V$ of dimension $n$ with inner product $g$ is a continuous map \[ P : \underbrace{V \times \cdots \times V}_r \to V \] satisfying

1) $P$ is skew;
\[ g\big( P(v_1, \ldots ,v_r),v_i\big) = 0 \ , \ 1 \leq i \leq r \ , \] 2) $P$ respects $g$;
\[ g\big(P(v_1, \ldots ,v_r),P(v_1, \ldots ,v_r)\big) = \det g(v_i,v_j) \ . \]

These were classified by Eckmann and Whitehead (see "George W. Whitehead, Note on cross-sections in Stiefel manifolds, Comment. Math. Helv.37 (1962/1963)") using algebro-topological methods. They were later also classified by Brown and Gray (see "Robert B. Brown and Alfred Gray, Vector cross products, Comment. Math. Helv. 42 (1967)") where those authors included an extra axiom: $P$ has to be multilinear. This extra axiom makes no significant difference to the classification. The classification theorem is:

An $r$-fold cross product on a real vector space $V^n$ exists if and only if we have one of

$\bullet$ $n$ even, $r=1$,
$\bullet$ $n=7$, $r=2$,
$\bullet$ $n=8$, $r=3$,
$\bullet$ $n$ arbitrary, $r=n-1$.

The proof of Brown and Gray uses Hurwitz' structure theorem for composition algebras. If you add a dimension to $V$ you can define a composition algebra, and vice versa. Their paper is my favourite reference. They actually consider a more general situation where the bilinear form is indefinite, that leads to more cross products (but only in the four cases above). They even work with any field of characteristic not $2$.

Using the standard inner product on $V = \mathbb{R}^n$ (which you are implicitly using by referring to $\ast$), your cross product is the last one on the list (it is a cross product in this sense, I was wrong to say otherwise earlier). It is normally just called the volume form, with the appropriate identifications made. I don't know about the earlier history of that particular case, or whether this was thought of as a cross product prior to the papers I've mentioned.

As Ryan points out, in two dimensions there is a $1$-fold cross product, and this is rotation by $90^{\circ}$. That's because $1$-fold cross products are the same as complex structures. The usual $2$-fold cross product on $\mathbb{R}^3$ is the volume form, and fits into the fourth case of the classification.

I cannot miss the chance to briefly mention the role of cross products in geometry. One can define a cross product on the tangent bundle of a Riemannian manifold. Kaehler manifolds are those with (parallel) $1$-fold cross products, and the cross products in seven and eight dimensions correspond to the exceptional holonomies.

Answer 2

This construction is in Spivak's book "Calculus on Manifolds".

Answer 3

I may be missing something, but what you have is simply a column of the adjoint matrix. Instead of formal vectors, put random entries into the first row of your matrix, call that $A.$ It has a determinant. The basic relation is that $A \mbox{adj}(A) = (\det A) I.$ That is, the "dot product" of any of rows $2$ through $n$ with the first column of $ \mbox{adj}(A)$ is $0.$

The first column of $ \mbox{adj}(A)$ is exactly what your formal determinant produces. This is a standard way to present the traditional cross product in basic physics classes.

http://en.wikipedia.org/wiki/Adjugate_matrix