new products in geometric algebra

From: Mark Adams (markjadams_at_lycos.com)
Date: 10/22/04 

Date: 22 Oct 2004 07:30:08 -0400

Hello,

I've been studying some of the additional operations defined in the
literature on "geometric algebra", AKA Clifford algebra. There are some
very interesting constructions, but I am having a hard time mapping them
to more common operations on the exterior algebra.

One question I have: has anyone come up with a clean definition in terms
of Clifford (geometric) multiplication for the "inner product" as
defined on the exterior algebra to define the Hodge star? For A and B
k-blades (i.e. the exterior products of k vectors A_i and B_j) this is
the construction

<A, B> := det(<A_i, B_j>)

with the result defined to be zero for blades of different grade
(composed of different numbers of vectors). Even an operation that
coincided with this for two k-blades would be of interest. I've looked
into various operations including:

 - "inner product" A.B := <AB>_|j-k| where A and B are j- and k-vectors
 - "commutator product" A x B := (AB - BA)/2
 - "contraction" A J B := (A /\ (Bw))w^-1 where w is the volume element
(pseudoscalar) and J is supposed to be a backwards "L"

but none seem to do the job.

A related item: I've been putting the definition of the Hodge star in
terms of geometric algebra operations

*A = ((w^-1)A)^+ where w^-1 is the inverse of the volume element
(pseudovector), and ^+ indicates reversion

into some more usable (for what I'm doing) forms. Can anyone verify
these results? Here A is a k-blade, w is the volume element
(pseudoscalar), and s is the number of negative signs in the signature
of the inner product defining the Clifford algebra.

*A = (-1)^(k(k-1)/2 + s) Aw
*A = (w^+w)(A^+)w

Thanks in advance for any and all help...

Relevant Pages

  • Re: This Weeks Finds in Mathematical Physics (Week 223)
    ... Gian-Carlo Rota ... > N.B. Geometric algebra shouldn't be confused with algebraic ... All users of applied mathematics should know something about this powerful ...
    (sci.physics.research)
  • PAUSH [This Weeks Finds in Mathematical Physics (Week 223)]
    ... > N.B. Geometric algebra shouldn't be confused with algebraic ... Groebner basis computations involving ideals in polynomial rings, ... All users of applied mathematics should know something about this powerful ...
    (sci.physics.research)
  • Re: This Weeks Finds in Mathematical Physics (Week 223)
    ... > When geometric algebra and linear algebra battle to win the hearts and ... because it's not even as if geometric algebra is ... > polynomial rings) in favor of learning about geometric algebra. ...
    (sci.physics.research)
  • Re: Geometric Algebra
    ... >> basis elements of a algebra for example. ... > I had a brief interest in geometric algebra when I stumbled upon ... > rotations for free. ... The quaternions are nicely embeddable inside ...
    (sci.physics.research)
  • Re: Programming languages for the very young
    ... > generalization of complex numbers, quaternions, octonions, ... > geometric algebra. ... Are you talking about Clifford Algebra or basic operator algebra over ...
    (comp.lang.lisp)