Is our current definition of the interior product standard?

Give it to me straight: is the thing called "interior product" on the
wiki the standard definition of the interior product, or is it a
simplified version for music theory? Do authors ever differ on this?

The more exterior algebra I learn, the less I'm ever sure if we're
using the same terms as anyone else.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Give it to me straight: is the thing called "interior product" on the
> wiki the standard definition of the interior product, or is it a
> simplified version for music theory? Do authors ever differ on this?

It would be more correct to say a definition one often sees, based on an inner product, is a dumbed-down version of what's on the wiki. Same comment for dual. What we have, without inner products, is right for our purposes.

On Wed, Nov 28, 2012 at 10:49 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>>
>> Give it to me straight: is the thing called "interior product" on the
>> wiki the standard definition of the interior product, or is it a
>> simplified version for music theory? Do authors ever differ on this?
>
> It would be more correct to say a definition one often sees, based on an
> inner product, is a dumbed-down version of what's on the wiki. Same comment
> for dual. What we have, without inner products, is right for our purposes.

Yes, I'm familiar with the way they dumb things down with inner
products. But just so I know how to communicate with other
mathematicians, if I post something on mathoverflow that uses the term
"interior product" exactly as we use it here, should I expect that
people familiar with exterior algebra will know what I'm talking
about? If not, what should I call it? Is there an established name for
it? (Do people even use exterior algebra like we do?)

I see tons of conflicting things about how the interior product is
defined. The Wikipedia page on "Exterior algebra" defines it just like
you do, but it only allows you to take the interior product of a
multivector and a grade-1 linear functional, so yours is more general.
But in Mathworld, they define it in terms of a binary operation
defined on /\(V) x /\(V), aka the thing I called the "regressive
product" a week or two ago (which for some reason is usually defined
using the Hodge star, even though you don't really need an inner
product to define it):

http://mathworld.wolfram.com/InteriorProduct.html

They also have a note at the bottom that basically states that if and
only if you accept an inner product on V do you get an isomorphism
between V and V*, at which point their interior product and your
interior product are the same.

Browne also defines it the same way as Mathworld, but that doesn't
mean much to me these days as I'm not a fan of the way Browne defines
things.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Yes, I'm familiar with the way they dumb things down with inner
> products. But just so I know how to communicate with other
> mathematicians, if I post something on mathoverflow that uses the term
> "interior product" exactly as we use it here, should I expect that
> people familiar with exterior algebra will know what I'm talking
> about? If not, what should I call it? Is there an established name for
> it? (Do people even use exterior algebra like we do?)

Call it the interior product but make it clear no inner product is defined and used.

I was first introduced to exterior algebra in a course on differential topology, and from a differential topology point of view, you distinguish V from V* and relate 1-forms to vector fields by means of the interior product, not inner product. When I first started posting about this stuff I relied on my memory because I couldn't find any good on-line references. It's still all messed up in my view.

> I see tons of conflicting things about how the interior product is
> defined.

It's a load of crap, really. I tried to explain it for us, in a good way.