Register to reply

Complexifying su(2) to get sl(2,C)-group thread footnote

by marcus
Tags: cgroup, complexifying, footnote
Share this thread:
marcus
#1
Aug10-03, 10:07 AM
Astronomy
Sci Advisor
PF Gold
marcus's Avatar
P: 23,101
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)
Phys.Org News Partner Mathematics news on Phys.org
'Moral victories' might spare you from losing again
Fair cake cutting gets its own algorithm
Effort to model Facebook yields key to famous math problem (and a prize)
Tyger
#2
Aug10-03, 11:40 AM
Tyger's Avatar
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?
r637h
#3
Aug12-03, 01:00 AM
P: n/a
Well, there you go: Topology/Non-Euclidian Geomerty, like poverty and ignorance: We will always have them with us.

Rudy

"Go Figure." - Archimedes


Register to reply

Related Discussions
Isometry sub-group 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 Astronomy & Astrophysics 118
Group Story thread General Discussion 13
Complexifying Lie algebras (footnote to group thread) General Math 2