John Stillwell - Naive Lie Theory.pdf - Index of

64 3 Generalized rotation groups
In SO(2m) we have the T m consisting of the matrices R θ1 ,θ 2 ,...,θ m
. In
SO(2m + 1) we have the T m consisting of the “padded” matrices
R ′ θ 1 ,θ 2 ,...,θ k
=
⎛
⎞
cosθ 1 −sinθ 1
sin θ 1 cosθ 1 cosθ 2 −sinθ 2
sinθ 2 cos θ 2 . .. .
⎜
cosθ k −sinθ k
⎟
⎝
sinθ k cosθ k
⎠
1
In U(n) we have the T n consisting of the matrices Z θ1 ,θ 2 ,...,θ n
.InSU(n) we
have the T n−1 consisting of the Z θ1 ,θ 2 ,...,θ n
with θ 1 + θ 2 + ···+ θ n = 0. The
latter matrices form a T n−1 because
⎛
e iθ ⎞ ⎛
1
e i(θ ⎞
1−θ n )
. .. ⎜
⎟
⎝ e iθ n−1 ⎠ = eiθ n
. .. ⎜
⎟
⎝
e i(θ n−1−θ n ) ⎠ ,
e iθ n
and the matrices on the right clearly form a T n−1 . Finally, in Sp(n) we
have the T n consisting of the matrices Q θ1 ,θ 2 ,...,θ n
.
We now show that these “obvious” tori are maximal. As with SO(3),
used as an illustration in the previous section, the proof in each case considers
a matrix A ∈ G that commutes with each member of the given torus
T, and shows that A ∈ T.
Maximal tori in generalized rotation groups. The tori listed above are
maximal in the corresponding groups.
Proof. Case (1): T m in SO(2m), form ≥ 2.
If we let e 1 ,e 2 ,...,e 2m denote the standard basis vectors for R 2m ,then
the typical member R θ1 ,θ 2 ,...,θ m
of T m is the product of the following plane
rotations, each of which fixes the basis vectors orthogonal to the plane:
rotation of the (e 1 ,e 2 )-plane through angle θ 1 ,
rotation of the (e 3 ,e 4 )-plane through angle θ 2 ,
.
rotation of the (e 2m−1 ,e 2m )-plane through angle θ m .
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