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

[Bertfried Fauser]

On Tue, 8 Jun 2004, Page, Bill wrote:

Dear Bill,

        I try to recover from my holidays and a conference, to get the
paper work off the desk, so this time only a medium size response.

> > > Formal definition
> > >
> > > ... bla bla ...
> > >
> > > such that the following diagram commutes:
> > >
> > >                V ----> C(q)
> > >                |     /
> > >                |    / Exists and is unique
> > >                |   /
> > >                v  v
> > >                A
> > >

Indeed, this construction is categorial, but not algorithmic. Hence almost
useless for computer algebra.


> > 2) The form should be a bilinear form, not only a quadratic
> > form. See
>
> I do not understand why you say these generalizations
> make the constructive definition any less obvious. It
> seems to me that we can have tensor algebra over a
> module

Look at the (anti)commutation of two grade one elements, its symmetric by
construction, so

        a (x) b + b (x) a = 2g(a,b)

has to have a symmetric bilinear form. To generalize this, you need an
_order relation_ on the grade one elements, to decide if you expand for
a (x) b (or for b (x) a), say we take lexicographic order and lets assume
e_i < e_j if i<j. Then one can generalize. However, one has to deal with
the effects with the linear ordering of teh bases (grade one space), this
is quite a peculiar and problemetic thing to do!

>   http://planetmath.org/encyclopedia/TensorAlgebra.html
>   http://planetmath.org/encyclopedia/CliffordAlgebra2.html

I will look for these, but hadn't yet time.


> "Let M be a (left-)module over a ring R, and B:VxV->k a
> symmetric bilinear form. Then the Clifford algebra Cliff(M,B)
> is the quotient of the tensor algebra T(M) by the relations
>   v (x) w + w (x) v = -2 B(v,w)"
>
> where (x) denotes tensor product, makes good sense to me.
> I don't see any problem if B(v,w) is degenerate. The
> Z_2-grading on the tensor algebra is still inherited in
> the same way.

In my eyes, the Clifford algebra is associated to a (graded) symmetric
algebra (which is already a quotient of teh tensor algebra) and not
directly to the tensor algebra, so the construction spezializes as:

Tensor algebra -> Graded Symmetric Algebra (ie Grassmann involved!)
 -> Clifford Algebra (symplectic case = Weyl algebra involved!)

> I do not think one needs to make assumptions about a basis
> in order to prove that the Clifford algebra is filtered
>
>   http://planetmath.org/encyclopedia/FilteredAlgebra.html
>
> and (most of?) the construction of the exterior algebra
>
>   http://planetmath.org/encyclopedia/ExteriorAlgebra.html
>
> goes through.

Point is, that in physics one deals not with the the plain algebraic
structures, but withaugmented ones. Eg, in QFT, one has to deal with
normal ordered and time ordered n-point functions. These for an algebra.
The normal odered field operators form a bicommutative Hopf algebra
(graded symmetric Hopf algebra). However, the normal and time odered
algebras are *isomorphic as Hopf algebras*!! However, they describe
totally different physical entities under the evaluation (counit) map

<0| : \psi_1\psi_2 ... \psi_n : |0> = \phi_n
<0| T(\psi_1\psi_2 ... \psi_n ) |0> = \tau_n

and the transition is an bijective map, the Wick expansion. Both algebras
differ _only_ in their filtration and can be distinguished, since they are
evaluated with respect to the _same_ physical vaccum, which is not
transformed! This is one of the major applications of Clifford algebra
structures in QFT and should be modeled in computer algebra, as was done
in Clifford/Bigebra.


> Chevalley's recursive construction is interesting since it
> simultaneously builds the graded structure and the quotient.

A couple of clifford packages were build upon these relations. However,
for fully symbolic computations, the recursion produces terms, which
cancel out in the next step of recursion, so the methos if by fare not
optimal. Example

 (a /\ b) ° c = (a ° b - g(a,b) ) ° c          # look at the g(a,b) term
              = a ° ( b /\ c + g(b,c)) - g(a,b) c
              = a _| (b /\ c + g(b,c)) + a /\ ( b /\ c + g(b,c)) - g(a,b) c
              = g(a,b) c - g(a,c) b + g(b,c) a - g(a,b) c + a /\ b /\ c
              = g(b,c) a - g(a,c) b + a /\ b /\ c

etc....

Hopf:
> It is not immediately clear to me how this approach is
> reflected in your definition below. What part is the Hopf
> algebra and what part is the twist?

Mail is probably not the right diection to write this down. If you are
interested, you might find the 2nd chapter of my hablitation usefull for
finding variouse definitions of Clifford algebras, the Hopf approch is
exactly the subject of this work (math.QA/0202059 and a concise short
explanation is found in the cookeville proceedings chapter
math-ph/0208018) More techical stuff can be found in (math-ph/0212031 and
math-ph/0212032, where explicite Clifford and Bigebra calculations are
shown) There you will see also that nonsymmetric bilinear forms are
mandatory. ;-))


> > Define a category "graded module"
> > Define the categories symmetric and exterior algebras over
> > such a module (this amounts to introduce super symmetric
> > multilinear algebra)
>
> I think that this is equivalent to the tensor algebra
> defined above, isn't it?

Nearly, its algraedy the quotione to the Grassmann (symmetric) algebra.

> > Define the category of reflexive spaces with inner product.
>
> I presume you mean something more general than the
> inner product?

Depends on what you mean by inner product. In fact there are quite a lot
different such products available, but only a special bilinear form
(namely one which is a Laplace 2-cocycle) plays an algorithioc role. See
my habilitation....


> > Define the Clifford algebra as a Functor which assigns to
> > every reflexive module a Clifford algebra.
>
> Aw, now that's the hard part isn't it? But I think it is
> the same problem as in the less general case. We need
> (somehow) to efficiently construct the appropriate quotient
> of the tensor algebra.

Yes, indeed. However, this should be done in a so sensible way, that we
can deal with many structures at the same time.

tensor -> graded symmetric = [Grassmann | Symmetric] -> Clifford | Weyl

In this way, the algebra of symmetric functions is also a Clifford (Hopf)
algebra, ;-)))

> I think I *might* know how to do this an incremental fashion
> in terms of a map (i.e. a "remember table") that defines the
> partition
>
>   T(M) ->> Cliff(M,B)

Thats not so difficult, however, in Maple we failed to compute the
multiplication table! Due to memory and time constraints. Maple 5 was not
able to compute the thing due to a 2^15 bit pointer restriction and maple
6 onwards was not able to do it because of memory constraints (I have 1GB
though) and the computation did not finish in reasonable time (weeks!)
Hence the product has to be evaluated on the spot. In Clifford we have
hash tables for already computed products on a low level (bilinear form
independent, since the form may be changed during computation)

> Isn't this logically part of in the tensor algebra and the
> Z_2-grading?

No, I think about super symmetric (multi) linear algebra. A very good
reference is

@INPROCEEDINGS{grosshans:rota:stein:1987a
    ,AUTHOR      = {Grosshans, {Frank D.} and Rota, {Gian-Carlo} and
Stein, {Joel A.}}
    ,BOOKTITLE   = {conference board of the mathematical sciences regional
conference series in mathematics, number 69}
    ,ID          = {13}
    ,PAGES       = {i--xxi,1--80}
    ,PUBLISHER   = {American Mathematical Society}
    ,TITLE       = {Invariant theory and superalgebras}
    ,YEAR        = {1987}
}

There you will notice the complexity of the problem.

A further remark:

The most impressive and compelling paper on Clifford algebra is the
following (pair, the first one is sufficient for almost all computer
algebra issues about the subject)

@ARTICLE{rota:stein:1994a
    ,AUTHOR      = {Rota, {Gian-Carlo} and Stein, {Joel A.}}
    ,ID          = {6}
    ,JOURNAL     = {Proc. Natl. Acad. Sci. USA}
    ,MONTH       = {December}
    ,PAGES       = {13057--13061}
    ,TITLE       = {Plethystic {H}opf algebras}
    ,VOLUME      = {91}
    ,YEAR        = {1994}
}
@ARTICLE{rota:stein:1994b
    ,AUTHOR      = {Rota, {Gian-Carlo} and Stein, {Joel A.}}
    ,ID          = {7}
    ,JOURNAL     = {Proc. Natl. Acad. Sci. USA}
    ,MONTH       = {December}
    ,PAGES       = {13062--13066}
    ,TITLE       = {Plethystic algebras and vector symmetric functions}
    ,VOLUME      = {91}
    ,YEAR        = {1994}
}

Rota and Stein there developed a theory of high abstraction (I try to
understand this paper since 1999 (Ixtapa conference) in an algorithmic
way but have not yet fully suceeded ....) You will notice however, that
the paper _is_ algorithmic in nature, as is the above cited work too!

To my best knowlegde, the three citations I made in this mail are the most
unique sources to a super symmetric multilinear algebra which contains
almost all algebra necessary for eg QFT. However, its also the most highly
abstracted thing (beside one further paper on the representation of
matroids, which I will not cite) which I know, so its the best approch for
an AXIOM implementation of the subject.


My Problem:

Still I am not able to envision a data structure complex enough to hold
the problem and simple enough not to make everything totally unmanagable.
A good start would be to write an AXIOM package for supersymmetric
bi-commutative Hopf algebras, if thats done, cliffordization is relatively
easy to implement.

My Abilities:

Right now, I am in posession of algorithms, which I can document
precicely, of the above described strutures. So if I would have a domain
(category) where the implementation part is missing, but types, etc were
defined, I could come up with highly efficient algorithms for the
implementation part.

Details later.

ciao
BF.

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