John Stillwell - Naive Lie Theory.pdf - Index of

56 3 Generalized rotation groups
Path-connectedness of SU(n)
We can prove that SU(n) is path-connected, along similar lines to the proof
for SO(n) in the previous section. The proof is again by induction on n,
but the case n = 2 now demands a little more thought. It is helpful to use
the complex exponential function e ix , which we take to equal cosx + isinx
by definition ( for now. ) (In Chapter 4 we study exponentiation in depth.)
α −β
Given in SU(2), first note that (α,β) is a unit vector in C 2 ,
β α
so α = ucosθ and β = vsinθ for some u,v in C with |u| = |v| = 1. This
means that u = e iφ and v = e iψ for some φ,ψ ∈ R.
It follows that
α(t)=e iφt cos θt, β(t)=e iψt sinθt, for 0 ≤ t ≤ 1,
gives a continuous path
SU(2) is path-connected.
( α(t) −β(t)
β(t) α(t)
)
from 1 to
( α −β
β α
)
in SU(2). Thus
Exercises
Actually, SU(2) is not the only special unitary group we have already met, though
the other one is less interesting.
3.3.1 What is SU(1)?
The following exercises verify that a linear transformation of C n , with matrix
A, preserves the Hermitian inner product (*) if and only if AA T = 1. They can be
proved by imitating the corresponding steps of the proof in Section 3.1.
3.3.2 Show that vectors form an orthonormal basis of C n if and only if their
conjugates form an orthonormal basis, where the conjugate of a vector
(u 1 ,u 2 ,...,u n ) is the vector (u 1 ,u 2 ,...,u n ).
3.3.3 Show that AA T = 1 if and only if the row vectors of A form an orthonormal
basis of C n .
3.3.4 Deduce from Exercises 3.3.2 and 3.3.3 that the column vectors of A form
an orthonormal basis.
3.3.5 Show that if A preserves the inner product (*) then the columns of A form
an orthonormal basis.
3.3.6 Show, conversely, that if the columns of A form an orthonormal basis, then
A preserves the inner product (*).