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

[Page, Bill]

Bertfried and Tim,

On Tuesday, June 01, 2004 6:56 PM you wrote:
> 
> ... but I still do not see how to do it categorial and in
> a full generality.
>       This week I am technically still in vacations and next 
> week partly on a conference, so I will have a fast internet
> connection only from pre-next week onwards. I will try to
> take this time to think about a start implementation and what
> it should look like.
> 
> Categorical:
> 
> From a categorical point of view, there is a functor from the
> spaces having a quadratic (bilinear not necessarily symmetric)
> form into the category of associative algebras. To characterize
> this functor would be the most difficult thing. But this is an
> unsolved math problem.
>

>From http://en.wikipedia.org/wiki/Clifford_algebra

Formal definition

Let V be a vector space over a field k, and q : V -> k a quadratic form on
V. The Clifford Algebra C(q) is a unital associative algebra over k together
with a linear map i : V -> C(q) defined by the following universal property:

for every associative algebra A over k with a linear map j : V -> A such
that for every v in V we have j(v)^2 = q(v)1 (where 1 denotes the
multiplicative identity of A), there is a unique algebra homomorphism

    f : C(q) ? A

such that the following diagram commutes:

               V ----> C(q)
               |     /
               |    / Exists and is unique
               |   /
               v  v
               A

i.e. such that fi = j.

---------

Definition by a universal property

  http://en.wikipedia.org/wiki/Universal_property

is a common method of category theory. Unfortunately it is
rather too abstract on usually non-constructive. Perhaps what
you had in mind is the notion of adjoint functors

  http://en.wikipedia.org/wiki/Adjoint_functors

But I think best approach is to start from the general
tensor algebra

  http://en.wikipedia.org/wiki/Tensor_algebra
  http://planetmath.org/encyclopedia/TensorAlgebra.html

I can imagine that the general construction of the tensor
algebra on a vector space should be quite natural in Axiom -
perhaps most of this is already present in Axiom. But then to
complete the general construction of a Clifford algebra we
need to be able to compute the quotient ring generated by
the quadratic form.

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

I am not so sure how one can do this in Axiom without
introducing a basis. In general we need to be able to
define an equivalence relation over the tensor expressions.

The following paper is rather difficult for me but 
seems interesting.

  http://www.nd.edu/~alhonors/pages/CliffordFinal.pdf

The Clifford Algebra in the Theory of Algebras, Quadratic
Forms, and Classical Groups by Alexander Hahn.

And the following work by Paul Leopardi also seems very
relevant.

  http://cage.rug.ac.be/~krauss/iciam.pdf

  http://www.maths.unsw.edu.au/~leopardi/

  http://www.maths.unsw.edu.au/~leopardi/clifford-2003-06-05.pdf

  http://sourceforge.net/projects/glucat/

Regards,
Bill Page.