[racket] SU(2) and friends

From: Gregory Woodhouse (gregwoodhouse at me.com)
Date: Fri Apr 22 15:35:52 EDT 2011

Clifford algebras (which are non-associative) are higher dimensional generalizations of H (the quaternions), so in the case of SU(2), you don't really need the machinery of Clifford algebras. That being said, I'm not entirely sure what you are looking for. Ordinary rotations (the circle group) can be embedded in SU(2) in various ways. Beyond that, you need a set of generators, such as the Pauli spin matrices (the chemistry connection). These slides


may be helpful there. Otherwise, the main thing to look at is whether representations are real or not (i.e., map group elements to real matrices). A complex rotation may look like dilation or a shear in real coordinates. Topologically, SU(2) is a double cover of SO(3) and the covering map p: SU(2) -> SO(3) associates two distinct complex rotations to each (3-dimensional) real rotation. That's where the split between fermions and bosons comes from.

On Apr 22, 2011, at 12:08 PM, Paul Ellis wrote:

> Many thanks for the continuing interest. 
> As you say, the 2nd is closer to my level and more accessible to me.
> (The first is likely to take me rather longer to be able to tackle.)
> Something  I came across in the late Pertti Lounesto's "Clifford
> Algebras and Spinors" (2e, 2001; ISBN 0-521-00551-5), which resembles
> your 2nd ref,  makes statements such as: 
> "In general, a rotation in R^4 has two invariant planes which are
> completely orthogonal, in particular they have only one point in
> common." ( p. 83)
> and 
> "There are three different kinds of rotations in four dimensions..." (p.
> 89).  
> Do you think there are any references to spaces of two complex
> dimensions that make statements of this sort? Or maybe I should be
> working them out for myself.

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