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

[Bertfried Fauser]

On Tue, 1 Jun 2004, Page, Bill wrote:

> Bertfried and Tim,

> 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.

Of course, this is standard! But....

1) Usually you will need Clifford algebras over rings, not fields, like
the ring of smooth functions -> tempered distributions

2) The form should be a bilinear form, not only a quadratic form. See

   symmetric forms =  bilinear forms mod altermating forms

iff char of teh base field/ring is not 2. The above construction is then
no longer so obvious.

3) ANY base introduces a filtration. Physics is _sensitive_ to this
filtration, even if the algebras are _isomorphic_ and hence mathematically
indistingishable. This stems from additional features employed in physics.
Hence it is _tremendously dangerous_ to introduce bases. However, certain
physical appliocations _need_ such a reference base, alas I am confused.

4) From a technical point of view, the most easy construction of a
Clifford algebra (and that one which is capable of arbitrary even
degenerated bilinear forms) is that of Chevalley, which unfortunately
assumes a grading, which is _not_ inherent in a Clifford structure.

Let x,y in V, u,v,w in V^, the exterior (Grassmann) algebra over V (unique
up to isomorphisms) define the Clifford product recursively as:

i) x _| y = B(x,y) = y |_ x    (bilinear for acting on VxV )

ii) x _| (u /\ v) = (x _| u) /\ v + ßhat{u} /\ (x _| v)   (derivation)

iii) u _| ( v _| w) = (u /\ v) _| w  (left action on the module V^)

definition:  the Clifford multiplication wrt the bilinaer form B (used in
the contraction _|) is defined as

  x ° u = x _| u + x /\ u

a general element v can be multiplied by recursive use of i) ii) and iii)

This is not the most efficient algorithm, but a plain approch. A much
better approch uses Hopf algebras and defines the Clifford product as a
twist by a (Laplace) 2-cocycle on the Grassmann Hopf algebra.

Hence the way to go is:

Define a category "graded module"
Define the categories symmetric and exterior algebras over such a module
(this amouts to introduce super symmetric multilinear algebra)
Define the category of reflexive spaces with inner product.
Define the Clifford algebra als a Funtor which assigns to every reflexive
module a Clifford algebra.

Much better is to make this that way that an itteration can be done.

It is mandatory that such modules may be direct sum of graded modules and
that one can somehow manage the grading information, eg we should think of
modules composes from symmetric and exterior parts

 M = Sym(V) \oplus Alt(W)

this is needed for more advanced Cliffordizations.

hope this helps
ciao
BF.

PS: Tim, with wiki I was refering to the wikipedia pages.

% 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>
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