Page 144 - Matrices theory and applications

A t C + 7.6. The Symplectic Group Sp n 127
x 2

C −

x 1
Figure 7.1. The Lorentz cone.

which is not contractible, while a convex set is always contractible. Since
M is a homeomorphism, one deduces that necessarily, MA = A, while
MC + = C ± , MC − = C ∓ .
The transformations that preserve C + , and therefore every connected
component of C,formthe orthochronous Lorentz group. Its elements are
T
those that send the vector e 0 := (1, 0, 0, 0) to C + ; that is, those for which
the first component of Me 0 is positive. Since this component is A (here it
is nothing but a scalar), this group must be G ++ ∪ G +− .

7.6 The Symplectic Group Sp n

Let us study first of all the maximal compact subgroup Sp ∩ O 2n .If
n

A B
M = ,
C D
with blocks of size n × n,then M ∈ Sp means that
n
T
T
T
T
T
T
A C = C A, A D − C B = I n , B D = D B,
while M ∈ O 2n yields
T
T
T
T
T
T
A A + C C = I n , B B + D D = I n , B A + D C =0 n .