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[Axiom-developer] Re: Clifford algebras in AXIOM

From: root
Subject: [Axiom-developer] Re: Clifford algebras in AXIOM
Date: Thu, 1 Apr 2004 07:33:30 -0500


I'm on vacation next week and aside from a weekend trip to visit my
mom I'll have some free time to think about this. I'll look at
providing some documentation about how to write a new domain in Axiom.

I'm also trying to come up to speed on your physics work. I have a 
textbook by Kaku (Kaku, Michio, "Quantum Field Theory" Oxford Univ. Press
(1993)). Chapter 2 deals with symmetries, SU(3) and SO(2), etc. My
efforts to understand that have led me off to more general Lie groups
(Lipkin, Harry, "Lie Groups for Pedestrians" Dover (1966)). Am I on 
the right track? If not, can you send me a reference or two?



>       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_
>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.

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