n-dimensional “cross product” reference request

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I guess I could just refer to her book, but was wondering if anyone knows a better/historical reference?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!

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If you do a web search with the dimension 1,3, and 7, you may find your cross product behaves as exepected only in those dimensions, as well as the references you desire. Gerhard "Ask Me About System Design" Paseman, 2012.04.17 – Gerhard Paseman Apr 17 '12 at 20:18
vol.90 (1983)The equality you mention is an immediate application of multi-linearity, most often left as an exercise to the reader. As for the Hodge * operator perhaps Hodge's book on harmonic integral's. As for the cross products in the exotic dimensions 1,3,7 i refer to the nice article by W. Massey, Amer. Math. Monthly, vol.90 (1983), 697-701, jstor.org/stable/2323537 – Liviu Nicolaescu Apr 17 '12 at 20:32
The product you have written above is the volume form of $\R^n$, and wouldn't be called a 'cross product' in the sense the term is usually used (except for $n=3$). I believe cross products on real vector spaces of arbitrary dimension were first defined by Eckmann in 1943. I can easily provide references for all the important papers on these (as I have done a good search before), but I'm not sure if that's what you're looking for? – Paul Reynolds Apr 17 '12 at 21:11
Thanks for the comments. I'll have to look up the paper cited by Liviu later. But from what I can tell you are all referring to a binary product (which only works in 1, 3, and 7 dimensions (a fact related to the existence of division algebras in 2, 4, and 8 dimensions). As for Paul's comment, what I have above really isn't the volume form (the standard volume form on $\mathbb{R}^n$ is the determinant) but instead this is something built from the volume form. As far as I can tell, Eckmann's work is again about the binary products in 1, 3, and 7 dimensions. – Bill Cook Apr 18 '12 at 0:43
I've always called it the cross product. Also, it's nice to point out what it means in $\mathbb R^2$ -- rotation by 90 degrees. But like you I don't know the appropriate historical reference. I imagine it was observed fairly early in the determinant literature. – Ryan Budney Apr 18 '12 at 4:16

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.

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Paul, thanks for the detailed answer! I will gladly accept it if you'll fix the classification statement. I believe you forgot to include $n=3$, $r=2$ (the classical cross-product) and $n=4$, $r=3$ in the list. – Bill Cook May 1 '12 at 13:29
Bill, you're welcome! In the paper of Brown and Gray the two cases you mention are written in the classification next to the seven and eight dimensional cases. However, they are actually already included in the fourth case. I prefer to write the classification as above to emphasise that the special cross products are really those for $n=7,8$ and not $n=3,7$ as often stated. I edited slighty to clarify. – Paul Reynolds May 2 '12 at 11:35
Of course, now I feel silly. Thanks Paul! – Bill Cook May 6 '12 at 20:00

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

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Pages 84-85 in chapter 4 "Integration on Chains". Thanks for the reference! – Bill Cook Apr 18 '12 at 18:54

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.