Talk:Abstract Algebra/Clifford Algebras

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[edit] Geometric Algebra link on this page.

I've been learning Geometric algebra and some associated physics as a self study project. I see there's a link on this page under see also:

http://en.wikibooks.org/w/index.php?title=Geometric_algebra&action=edit&redlink=1

but this page does not exist. Was it intended to be a chapter of this abstract algebra book or a separate book? I've dumped a fair amount of simple geometric algebra content into the wikipedia page:

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

but not all of it is really appropriate (I have the urge to write more than is encyclopedic, and have started to collect notes in standalone latex instead of contributing to wikipedia). I'd say that it would make more sense to move some of it here, but perhaps as a separate (physics biased) book.

A very rough possible outline is as follows:

motivation and definition
   dot product
   wedge product
axiomatic development
   contraction, linearity, associativity
identities
   norm
   vector inversion
   lagrange.
   determinants
   grade
   reversion
   generalized dot and wedge product
   commutator
examples
   orthogonal decomposition
   bivectors
   trivectors
   generalized exponentials
   cross product
   complex numbers
   quaternions
   reciprocal frame relations
      raised and lowered indexes
      components of vectors with non-orthonormal basis vectors.
   gradient
   antisymetric tensors and bivector representation.
   reduction identities
   wedge product use to solve linear systems.
geometry
   lines, planes, conics, ...
   angle between lines, planes, ...
   projection and rejection
      comparision to matrix methods.
   rotors
   area
   volume
   intersection of spaces
      line and line
      line and plane
      plane and plane
physical examples
   torque
   radial velocity and acceleration
   spacetime gradient
   lorentz boost
   maxwells equations
   rigid bodies
      kinetic energy of rotating body.  comparision to cross product variation.
   spherical triangulation
      (measurement of angles from well separated points on the earth).
history
references and learning material

Peeter.joot (talk) 17:32, 1 August 2008 (UTC)

[edit] Use of α in the definition of the Clifford group

Many authors define the Clifford group slightly differently, by replacing

the action xvα(x)−1 by xvx−1. This produces the same Clifford group.

I believe the statement that "this produces the same Clifford group" is wrong. Or I'm confused. Consider the case of a positive definite quadratic form Q over a 3-dimensional real vector space V, and write e1,e2,e3 for some orthonormal basis. Then the element u = e1e2e3 of the Clifford algebra is central and its inverse is − u which is also α(u) and ut. Being central, it belongs to the Clifford group with either definition: so far so good. But now consider the element x = \cos\theta + \sin\theta\,u: this is again central, so that xvx − 1 = v for all v in V and it belongs to the Clifford group with the second (alternative) definition. However, α(x) − 1 = x so that xv\alpha(x)^{-1} = x^2 v = \cos2\theta\, v + \sin2\theta\,uv and the second term is not in V for θ not a multiple of π/2 so that x does not belong to the Clifford group with the first (main) definition. Also, this element x is neither purely even nor purely odd. Presumably (actually, quite trivially) the definitions become equivalent if x is assumed up front to be purely even or odd, but the one with xvα(x) − 1 does not need to assume this (or something). Any comments before I try to fix this? --Gro-Tsen (talk) 18:49, 4 September 2009 (UTC)

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