Random stuff about exterior algebra and interior products

Gene said that the situation with exterior product terminology is
messed up. I've made sense of what's going on and found a simple way
to un-mess it up. TL;DR at the end.

Exterior algebra turns up in a lot of places. Of those places, the
people who think of things closest to the way that we do are the
"geometric algebra" people, where geometric algebra is really the term
for any sort of Clifford algebra over the space of the reals. The
downside is that they a) use goofy terminology, and b) require a
quadratic form which usually defines an inner product (which doesn't
have to be positive definite). The upside is that a) it's simple (and
informative) to translate their terminology into standard terminology,
and b) I've found that by using Gene's dual instead of the Hodge dual,
and that by using the dual space rather than an inner product, many
concepts in geometric algebra can be translated into more general
concepts in exterior algebra.

The Wikipedia article[1] references Dorst[2] who defines and
identifies several types of product related to this inner product,
such as the "left and right contractions," the "fat dot product," and
"Hestenes' inner product." (Ironically, the thing that Hestenes[3]
actually calls the "dot product" is actually the "fat dot product,"
but whatever.)

Looking a bit closer reveals that their "fat dot product" is basically
Gene's "interior product, with a "twist": they've defined it in an
ass-backwards way using the geometric inner product and Hodge duality,
as Gene has forewarned us about. So instead of it being something
between a multivector and an element of the dual space, it's now
something between two multivectors. However, it's fairly simple to
redefine this product using linear functionals and Gene's duality
instead of inner products and Hodge duality. If you make this
adjustment, you get exactly Gene's interior product.

Dorst's two contractions can be generalized in the same way, but they
don't seem to be all that useful. The left contraction of a val and a
bimonzo is a monzo, but the left contraction of a bival and a monzo is
zero. The opposite applies to the right contraction. As is the case
with the "fat dot product," Gene's interior product is the version
which combines the two so that you don't ever get a zero result, much
like the "fat dot product," which is rather useful in that regard.

The same applies to Dorst's and Hestenes "scalar product." The scalar
product basically envisions any mixed-grade multivector as a single
vector; likewise with any mixed-grade element in the dual space, which
becomes a single linear functional; the two are then applied to one
another. For example, if m = a + |b c> + ||d>> and v = e + |f g> +
||h>>, then m * v = a*e + b*f + c*g + d*h, where * is the scalar
product. This does seem useful when working with mixed-grade
multivectors.

Since these are rather simple and fundamental geometric algebra
operators, there are lots of theorems involving them. Once you've made
the above adjustments to these operators, you can then proceed to
continue looking at Hestenes' and Dorst's research and seeing how
these adjustments interact with their theorems. The above will
probably be more useful if we ever start using mixed-grade
multivectors for any reason; these have already come up while studying
Cangwu badness. And for instance, it doesn't really matter if Dorst's
two products end up having any use (you can read his paper and decide
if they do or not); the point is that they can easily be translated
into the language of what we're doing. The real point is that this
approach applies to other concepts in geometric algebra, which I've
found can often be translated into simpler concepts that apply to
exterior algebra in general by replacing Hodge duality with Gene's
duality and inner products with the dual space.

Lastly, it's notable that the "inner product" of geometric algebra
need not be positive definite, unlike actual inner products; this is
the case with the Minkowski inner product from physics, for instance.
It also only exists in certain "non-degenerate" cases, the details of
which we don't need to worry about. What is certain is that the usual
Euclidean inner product does indeed imply the quadratic form necessary
to define a Clifford algebra. Therefore, whenever we're working with
something like the TE norm, we've naturally gained the structure of a
real Clifford algebra, so all of the theorems of geometric algebra
apply to what we're doing.

-Mike

[1] - http://en.wikipedia.org/wiki/Geometric_algebra#Extensions_of_the_inner_and_outer_products
[2] - http://staff.science.uva.nl/~leo/clifford/inner.ps
[3] - http://geocalc.clas.asu.edu/pdf/UGA.pdf

On Tue, Dec 11, 2012 at 5:10 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
> Gene said that the situation with exterior product terminology is
> messed up. I've made sense of what's going on and found a simple way
> to un-mess it up. TL;DR at the end.

Oops, forgot the TL;DR. Here it is:

TL;DR: different authors use exterior algebra in completely different
contexts, sometimes each with slightly different definitions of things
and with their own context-specific terminology. You can "translate"
many geometric algebra definitions into our language by using Gene's
duality and dual spaces rather than Hodge duality and inner products,
yielding more general and useful concepts; you can also translate
their strange terminology into normal terminology (i.e. "outer
product" -> "wedge product"). After such translation, it's probably
true that the geometric algebra people are thinking about exterior
algebra in a way closest to what we're doing here, as opposed to
something like differential topology.

Also, whenever you're working with the TE norm, note that your
exterior algebra now naturally gains the structure of a real Clifford
algebra. Therefore, you now have a well-defined geometric product and
all of the theorems of geometric algebra suddenly apply miraculously
to music theory.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Gene said that the situation with exterior product terminology is
> messed up. I've made sense of what's going on and found a simple way
> to un-mess it up. TL;DR at the end.
>
> Exterior algebra turns up in a lot of places. Of those places, the
> people who think of things closest to the way that we do are the
> "geometric algebra" people, where geometric algebra is really the term
> for any sort of Clifford algebra over the space of the reals.

Can you spell out for us why we want to put things in a Clifford algebra setting? I thought about it some years back, but didn't see it did us any good.

On Dec 12, 2012, at 12:23 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:

Can you spell out for us why we want to put things in a Clifford algebra
setting?

We don't. The point of my post was to suggest that the converse is useful.

-Mike