Complexifying su(2) to get sl(2,C)group thread footnoteby marcus Tags: cgroup, complexifying, footnote 

#1
Aug1003, 10:07 AM

Astronomy
Sci Advisor
PF Gold
P: 22,803

On the group thread midterm exam (which we never had to take!) it says what is the LA of the matrix group SL(2, C)
and the answer is the TRACE ZERO 2x2 matrices. So that is what sl(2,C) is. When you exponentiate one of the little critters, det = exp trace, so the determinant is one which is what SL means. Any X in sl(2,C) has a unique decomposition into skew hermitians that goes like this X = (X  X*)/2 + i(X + X*)/2i and these two skew hermitians (X  X*)/2 and (X + X*)/2i are trace zero, because trace is linear check the skew hermitiandom of them: (X  X*)* = (X*  X) =  (X  X*) the other one checks because (1/2i)* =  (1/2i) since conjugation does not change (X + X*)* = (X + X*) so the upshot is that any X in sl(2,C) is composed X = A + iB of two matrices A and B in su(2) Also on the midterm was the fact that su(2) is the skew hermitian ones: A* =  A. There was this footnote on complexification of LAs and the above suffices to show, without much further ado, that su(2)_{C} the complexification of su(2) is isomorphic to sl(2, C) 



#2
Aug1003, 11:40 AM

P: 402

SL(2,C) is a representation of the group of boosts and turns, so why doesn't it show up in our descriptions instead of the 4×4 Dirac spinors?



#3
Aug1203, 01:00 AM

P: n/a

Well, there you go: Topology/NonEuclidian Geomerty, like poverty and ignorance: We will always have them with us.
Rudy "Go Figure."  Archimedes 


Register to reply 
Related Discussions  
Isometry subgroup of the gauge group & the center of structural group  Differential Geometry  0  
Lorentz group, Poincaré group and conformal group  Special & General Relativity  12  
The thread thread: Strangeness of the expanding space paradigm  General Astronomy  118  
Group Story thread  General Discussion  13  
Complexifying Lie algebras (footnote to group thread)  General Math  2 