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

  • Thread starter marcus
  • Start date
  • #1
marcus
Science Advisor
Gold Member
Dearly Missed
24,770
791
Complexifying su(2) to get sl(2,C)---group thread footnote

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)
 

Answers and Replies

  • #2
Tyger
398
0
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
Well, there you go: Topology/Non-Euclidian Geomerty, like poverty and ignorance: We will always have them with us.

Rudy

"Go Figure." - Archimedes
 
Last edited by a moderator:

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

Replies
4
Views
277
Replies
4
Views
166
Replies
9
Views
354
Replies
8
Views
468
Replies
1
Views
283
Replies
10
Views
482
  • Last Post
Replies
4
Views
427
  • Last Post
Replies
4
Views
575
  • Last Post
Replies
2
Views
305
  • Last Post
Replies
5
Views
471
Top