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## Re: [Axiom-developer] Re: Clifford algebras in AXIOM

 From: Camm Maguire Subject: Re: [Axiom-developer] Re: Clifford algebras in AXIOM Date: 01 Apr 2004 16:43:26 -0500 User-agent: Gnus/5.09 (Gnus v5.9.0) Emacs/21.2

Greetings!  For what its worth, I have some background in this area.
My Ph.D. is in theoretical physics, specifically the quantum theory of
solids.  I'm also currently involved in a project making some
significant use of group theory.  Unfortunately, I don't know anything
about axiom (yet) above the lisp level!

Take care,

> Bertfried,
>
> I'm on vacation next week and aside from a weekend trip to visit my
> 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?
>
> 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.
> >
>
>
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