[racket] Query for Gregory Woodhouse

From: Gregory Woodhouse (gregwoodhouse at me.com)
Date: Tue Apr 19 13:01:01 EDT 2011

If you're looking for a mathematical treatment of the representation theory of Lie groups, but need background on representation theory of finite groups (not a logical prerequisite, but pedagogically very helpful), you might look up William Fulton and Joe Harris, "Representation Theory: A First Course", GTM 129, Springer-Verlag, 1991 (ISBN 0-387-97495-4). Beyond that, I'd suggest starting with a good book on the representation theory of semisimple Lie algebras (N.B.) and then moving on to group representations. On the other hand, the representation theory of compact Lie groups (and this includes SU(n)) is more accessible because such groups have a group invariant finite measure known as the Haar measure. Integration with respect to this measure can replace the counting arguments used in the finite case and, personally, I find this more accessible than the strictly algebraic approach based on Lie algebras. At least the basic decomposition theorems are much easier to derive, and you can get into the meat of the subject more quickly without having to deal at the outset with technical results that have nothing to do with the classification theory. On the other hand, if all you're interested in is SU(2) an geometry in two (complex) dimensions, Fulton/Harris will give you what you need and more. In fact, in this case, you can work directly with H (the quaternions) and get the basic results you need pretty quickly.

Feel free to write to me off-line.

On Apr 18, 2011, at 1:17 AM, Paul Ellis wrote:

> Dear  Gregory Woodhouse
> Unrelated to Racket – I’m interested in the Geometry of 2 Complex Dimensions (“G2CDs”) and came across your 2006 post:
> [plt-scheme] 3rd-8th Grade
> Gregory Woodhouse gregory.woodhouse at sbcglobal.net 
> Mon Mar 20 08:51:15 EST 2006
> As it appears that you know something about the G2CDs, would you be able to point me towards any fairly elementary book/article that discusses it?
> I’m an ex-chemist (but not a mathematician) trying to obtain some intuitive insight into spinors, and came across Bruce Schumm’s “Deep Down Things”  (a popular level explanation of gauge theory) in which he starts discussing rotations in 2CDs. Rather than limiting it to the usual discussion of the Lie group SU(2), he does start to explain it in a bit more detail, and I’d very much like to pursue this, but he gives no references. I’m well aware of the common references to spinors, quaternions, Clifford algebras, and geometric algebra, but with one very limited exception, haven’t found any description, at the level of coordinates etc.,  of the type I seek.
> I would be most grateful for any constructive advice  
> Paul Ellis

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