Talk:Abstract Algebra/Clifford Algebras

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)

Use of α in the definition of the Clifford group

 * Many authors define the Clifford group slightly differently, by replacing

the action xvα(x)&minus;1 by xvx&minus;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 $$e_1,e_2,e_3$$ for some orthonormal basis. Then the element $$u = e_1 e_2 e_3$$ of the Clifford algebra is central and its inverse is $$-u$$ which is also $$\alpha(u)$$ and $$u^t$$. 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, $$\alpha(x)^{-1} = x$$ so that $$xv\alpha(x)^{-1} = x^2 v = \cos2\theta\, v + \sin2\theta\,uv$$ and the second term is not in V for &theta; not a multiple of &pi;/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 $$xv\alpha(x)^{-1}$$ 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)