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## [Axiom-developer] Clifford algebras in AXIOM

 From: Bertfried Fauser Subject: [Axiom-developer] Clifford algebras in AXIOM Date: Thu, 1 Apr 2004 12:15:01 +0200 (CEST)

Dear Tim,

I have had a look at the clifford package in AXIOM by now. Indeed,
it looks to me, as if this package was only included for the sake of
showing how to program, and not for really doing research in clifford
algebra. In effect, only product, sum, ...  are defined nothing special.
Furthermore is the restriction to a quadratic form not quite necessary.
I made good progress with some math to remedy this, but I have
problems to start to write a category and/or a domain (I tried it for an
afternoon or so, but nothing compiled, ....). As you announced to provide
a template with explanations, I would benefit very much from this.  At the
moment I do experiment with the Hopf algebra of symmetric functions, but
if that works, its easy to implement a super Hopf algebra of super
symmetric functions which includes symmetric and grassmann as weyl and
clifford algebras. My main problem is to define data structures in an most
general way, but such that it makes still sense mathematically (and is
not totally inefficient either, computations will be _tremendously_
involved).

I would need somethinng like:

Super [L]
* L is an "alphabet of letters, which may be signed letters,
there are positive, neutral, and negative letters, and later on
their 'duals'
* Letters may form words by concatenation  "letter monomials"
* words may be added to form general elements
sentences or "polynomials"
* Note that this is the tensor algebra over [L] factoered by
the symmetry constrained induced by teh letters

Tens Super [L]
* Now we itterate! Consider _any_ monomial in Super [L] as a
generator of Supper [L], denote it somehow eg (w)
* Create (as for super) the tensor algebra over the (w)'s
* Introduce some fancy maps and scalar products, build a factor
wrt to those, then one gets
Pleth Super [L] and its graded dual Brace Super [L]
* Pleth Super [L] has as special cases the symmetric functions,
the Grassmann algebra, Weyl and Clifford algebras, but also
Heisenberg algebras etc, exactly my beloved pets.

__Pleth Super [L] __ is a Hopf algebra!

Example: let L={a} be the alphabet in one neutral letter (ordinary
variable), then Super [L] becomes the polynomial algebra Z[a] respectivley
the algebra of formal power series Z[[A]] (all tensor products are taken
over the intergers Z). A monoidal basis of Super [L] in this case is given
by the powers of a, {1,a,a^2,a^3,a^4,...}, note that AXIOM has to type
these as "monom Letter neutral" (one should be able to distinguish between
monoms and polynoms for sake of algorithmical simplicity), then Tens Super[L]
is giben by the structure V = {1,a^2,a^3,a^4,...} and V^2, V^3 etc are
formed as linear combinations of products of these terms, eg
W \in Tens Super [L] with
W = (w1)(w2)...(wn)
(wi) = (a^r)^{s}(a^p)^(q)

hence something like:

x0 (1)+ x1 (1)^(4) + x2 (a)^(1)(a^2)^(3) + ....

there are still more complicated data structures to come, namely a letter
place algebra, where elements are formed from pairs of monomials of the
Tens Super [L] and Tens Super [P].

Of course one can discard letter place techniques, and call everything a
"module", The above described mechanism is in a certain sense basis free,
in the module language this become apparent.

Now, I searched for some time to see how AXIOM handles polynomials, and
variouse series, but wasn't really happy about the code, I simply don't
undertsand whats going on there. I suspect, that much of the code could be
reused for my problem, but the grading and letter type has to be added
carefully. I do by now know the algorithms how to compute for many
coercions etc, and would like to start with some baby category to see if I
am right.
An (simple) example how to define a category/domain and how to
define types (data structures) (and pretyprint output, otherwise these
data cannot be recognized by a human, I am thinking for a better notation
though, hopefully streamlined for applications in physice) would be of
extraordinary help.

cheers
BF.

% PD Dr Bertfried Fauser
%       Institution: Max Planck Institut for Mathematics Leipzig
<http://www.mis.mpg.de>
%       Privat Docent: University of Konstanz, Physics Dept
<http://www.uni-konstanz.de>
% contact |->    URL : http://clifford.physik.uni-konstanz.de/~fauser/