RE: [Axiom-developer] Clifford algebras in AXIOM

[Bertfried Fauser]

On Fri, 2 Apr 2004, Mike Thomas wrote:

Dear Mike,

perhaps this topic should move to teh AXIOM math mailing list? But I
provide an short answer, to others I beg your pardon.

> Your post caught my eye because I'm interested in the so called Geometric
> Algebra (GA) as promoted by David Hestenes eg:
>
>   http://modelingnts.la.asu.edu/
I know this flavour of Clifford algebra very well. Unfortunately it does
only serve en engineer, not a scientist. Clifford algebras were (almost)
classified by Clifford himself, virtually _anything_ is know and from a
mathematical point of view this field is dead. However, quaternions etc,
are used in computer graphics, image perception, robotics, etc.

What bothers me is the structure of quantum field theory, but under an
categorial angle. There one needs quite elaborated algebraic structures,
among them those which I described in my mail.

> At one stage I considered implementing a GA library in Haskell, a non-strict
> functional programming language.  My idea was that you would implement a
> general GA data type as a recursive lazy datastructure which would unfold
> during computations only to the extent needed, hopefully saving work -
> something I quite like to do!

I had a look at haskel, but decided that a full grown computer algebra
system provides such a larg amount of pre-done things, that I wont miss
it.

>   1. you were asking about data structures,

Indeed, I know the mathematical needs, but had to lear that representing
math in a computer sets further sever constraints, if perfomance is an
issue.

>   2. you seem to be constructing strings from symbols and constructing
> algebra on top of algebra, on top of algebra..., and

No, not strings, better to think of lists of lists of typed (lists,
elements, grades, signs,...)

>   3. GA seems to be a small subset of your algebraic structures,
Rather small yes.

        ndeed streams would be perfectly good for creating bases, etc, but
I need also to implement combinatorially very involved "products".
Consider the composition of functions. F a polynom, g a polynom, compute
(f\circ g)(x)=f(g(x)), this is already involved, now consider the
composition of f with u in Tens Tens V, (releted to products of group
representations)

> stream [a, f(a), f(f(a)), f(f(f(a))) . . .]. Given that the function
Usually to slow, itteration is computational inefficient (at least if I
handle it)

> worthwhile finding out.  It might also be worthwhile contacting Hestenes
> himself who is, I believe, looking into higher level computational methods
> for GA.

I know him personally, and especially Hongbo Li a post doc of him who did
some work in theorem proving. But my problem is much more complicated by
magnitutes. You might have a look at the Maple packages Clifford and
Bigebra (math.tntech.edu/rafal/) which Rafal Ablamowicz and myself
maintain. This can do almost all with tensor products of clifford
algebras. If I am looking for a new implementation in (either maple or
axiom) I will not make the errore you find documented there once more.

I stop since this is a developer mailing list ;-))
ciao
BF.

% PD Dr Bertfried Fauser
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