Re: [Axiom-developer] RE: learning in public

[Bertfried Fauser]

On Wed, 9 Jun 2004, William Sit wrote:

> QFORM    QuadraticForm in clifford.spad
> CLIF     Clifford Algebra in clifford.spad
>          (over any field K and quadratic form on K^n)
> GRMOD    Graded Module in carten.spad
> GRALG    Graded Algebra in carten.spad
> CARTEN   CartesianTensor in carten.spad (tensor algebra)
> CARTEN2  CartesianTensorFunction2 in carten.spad
> SYMFUNC  SymmetricFunctions in efstruc.spad (by Manuel Bronstein)
> PRTITION Partition in prtition.spd (by William Burge)
> SYMPOLY  SymmetryPolynomial in prtition.spad (by Wiliam Burge)
> BMODULE  BiModule in catdef.spad (probably by Watt also)

Dear Prof Sit,

        thats a good starting point. I was able only of the sym functions
and clifford/quadratic forms categories. I will try to understand whats
going on there. Perhaps I can "morph" some code into what I want to do.

> On the contrary, such universal objects are the key to construction of free
> objects and morphisms in the category (examples are vector spaces where you 
> can
> define morphisms by specifying only values on a basis; but also works for 
> higher
> algebras such as free shuffle algebra) and the non-free ones are done by 
> forming
> quotients.

Indeed, the value of universal constructions is beyond any doubt, but
they usually are not algorithmical. So not useful for implementations.

> To have different order relation or grading (filtration), it would be best to
> modify the source to allow an additional parameter or two. For example, in
> polynomial algebra, different term ordering can be done in GDMP (Generalized
> Distributed Multivariate Polynomial), which is a domain constructor by 
> modifying
> DMP or HDMP. A CliffordAlgebra CATEGORY is needed if the REPRESENTATION of the
> SAME (mathematically) algebra changes (or when the code for the basic 
> operations
> will be changed; for example if there are more efficient algortihms when the
> points are ordered). For polynomial algebra, a category is created so that
> different representations (DMP, POLY, SMP) can be used.

OK, that are further good hints where to look for a starting point.

> I suggest starting with generalizing the present CliffordAlgebra to include in
> order:
> (1) arbitrary symmetric bilinear form (may need SymmetricAlgebra, but may be 
> not
> by using SYMPOLY)
> (2) grading (and/or order) (using GRALG and GRMOD)
> (3) bimodule structure using BIMODULE
>
> Then when needed, create Hopf Algebra based on this experience.

Unfortunately, I see it rather the other way arround. If I need clifford
stuff, I can use our Clifford/Bigebra package. So doing a new atack to the
problem, I will try to do something rather fundamental, otherwise its not
woth the time effort, since I am computationally on a large subset. The
same point prevents me from trying "simply" to do an impementation of
Clifford/Bigebra in AXIOM.

My colleague, Rafal Ablamowicz, is using windows, so I will probably
benefit from a windows port and see with great pleasure that there is big
progress.

So, the name of the game is to write a (Hopf algebra powered) super
symmetric multilinear algebra package, which allows to itereate itself.
This includes:
* Grassmann and Symmetric algebras (superalgebras) [these are naturally Hopf]
* Symmetric functions [see Rota/Stein Plethystics algebras as cited in the
  previous mail and math-ph/0308043
* implementation of a twist deformation of these structures
* further functionality, like antipode, twists, etc
* representations
* ....

I will try to make a start :-)
ciao
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/
%              Phone : Leipzig +49 341 9959 735  Konstanz +49 7531 693491