John Stillwell - Naive Lie Theory.pdf - Index of

60 3 Generalized rotation groups
One can prove Sp(n) is path-connected by an argument like that used for
SU(n) in the previous section. First prove path-connectedness of Sp(2) as for
SU(2), using a result from Section 4.2 that each unit quaternion is the exponential
of a pure imaginary quaternion.
3.4.6 Deduce from the path-connectedness of Sp(n) that det(B 2n )=1.
This is why there is no “special symplectic group”—the matrices in the symplectic
group already have determinant 1, under a sensible interpretation of determinant.
3.5 Maximal tori and centers
The main key to understanding the structure of a Lie group G is its maximal
torus, a (not generally unique) maximal subgroup isomorphic to
T k = S 1 × S 1 ×···×S 1
(k-fold Cartesian product)
contained in G. The group T k is called a torus because it generalizes the
ordinary torus T 2 = S 1 ×S 1 . An obvious example is the group SO(2)=S 1 ,
which is its own maximal torus. For the other groups SO(n), not to mention
SU(n) and Sp(n), maximal tori are not so obvious, though we will find
them by elementary means in the next section. To illustrate the kind of
argument involved we first look at the case of SO(3).
Maximal torus of SO(3)
If we view SO(3) as the rotation group of R 3 ,andlete 1 , e 2 ,ande 3 be the
standard basis vectors, then the matrices
⎛
⎞
cosθ −sinθ 0
R ′ θ = ⎝sinθ cosθ 0⎠
0 0 1
form an obvious T 1 = S 1 in SO(3). The matrices R ′ θ
are simply rotations
of the (e 1 ,e 2 )-plane through angle θ, which leave the e 3 -axis fixed.
If T is any torus in G that contains this T 1 then, since any torus is
abelian, any A ∈ T commutes with all R ′ θ ∈ T1 . We will show that if
AR ′ θ = R ′ θ A for all R ′ θ ∈ T 1 (*)
then A ∈ T 1 ,soT = T 1 and hence T 1 is maximal. It suffices to show that
A(e 1 ), A(e 2 ) ∈ (e 1 ,e 2 )-plane,