Page 143 - Matrices theory and applications

7. Exponential of a Matrix, Polar Decomposition, and Classical Groups
126
Remark: σ admits a right inverse, namely
α
α → M := diag(α 1 1, 1,... , 1,α 21).
The group O(p, q) appears, therefore, as the semidirect product of G ++
2
with (ZZ/2ZZ) .
We deduce immediately from the proposition that O(p, q) possesses five
open and closed normal subgroups, the preimages of the five subgroups of
2
(ZZ/2ZZ) :
• O(p, q)itself;
• G ++ , which we also denote by G 0 (see Exercise 21), the connected
component of the unit element I n ,
• G ++ ∪ G α , for the three other choices of an element α.
One of these groups, namely G ++ ∪ G −− is equal to the kernel SO(p, q)
of the homomorphism M → det M. In fact, this kernel is open and closed,
thus is the union of connected components of O(p, q). However the sign of
det M for M ∈ G α is that of α 1 α 2 , which can be seen directly from the
α
case of diagonal matrices M .
7.5.2 The Lorentz Group O(1, 3)
If p =1 and q = 3, the group O(1, 3) is isomorphic to the orthogonal
2
2
2
2
group of the Lorentz quadratic form dt − dx − dx − dx , which defines
1
3
2
1
the space-time distance in special relativity. Each element M of O(1, 3)
corresponds to the transformation

t, t,
→ M ,
x x
which we still denote by M, by abuse of notation. This transformation
2
2
2
preserve the light cone of equation t − x − x − x 2 3 = 0. Since it is
1
2
4
a homeomorphism of IR , it permutes the connected components of the
complement C of that cone. There are three such components (see Figure
7.1):
• the convex set C + := {(t, x) | x <t};
• the convex set C − := {(t, x) | x < −t};
• the “ring” A := {(t, x) ||t| < x }.
Clearly, C + and C − are homeomorphic. For example, they are so via the
time reversal t →−t. However, they are not homeomorphic to A, because
2
2
2
the latter is homeomorphic to S × IR (here, S denotes the unit sphere),
1
We have selected a system of units in which the speed of light equals one.