Student Seminar: Classical and Quantum Integrable Systems

It is clear that
( ) 1 0
α + β
0 1
( ) 0 1
, α
−1 0
( ) 1 0
+ (β + 2πk)
0 1
( ) 0 1
−1 0
has the the same image under the exponential map. In the sufficiently small neighbourhood
of 0 in M(n, R) the map exp A is a diffeomorphism. The inverse map is
constructed by means of series
for x sufficiently close to the identity.
ln x = (x − I) − 1 2 (x − I)2 + 1 3 (x − I)3 − · · ·
Linear representation of a Lie algebra. Adjoint representation. Let J be a
Lie algebra. We say that a map
ρ : J → M(n, R)
defines a representation of the Lie algebra J is the following equality is satisfied
for any two vectors ζ, η ∈ J .
ρ[ζ, η] = [ρ(η), ρ(ζ)]
Let F (t) be a one-parameter subgroup in G. Then g → F gF −1 generates a oneparameter
group of transformations in the Lie algebra
AdF (t) : T → T .
The vector d AdF (t)| dt t=0 lies in the Lie algebra. Let a ∈ T and let F (t) = exp(bt)
then
d
dt AdF (t)| t=0 a = d (
)
exp(bt)a exp(−bt) | t=0 = [b, a]
dt
Thus to any element b ∈ J we associate an operator ad b which acts on the Lie
algebra:
ad b : J → J , ad b a = [b, a] .
This action defines a representation of the Lie algebra on itself. This representation
is called adjoint. To see that this is indeed representation we have to show that it
preserves the commutation relations, i.e. that from [x, y] = z it follows that
We compute
[adx, ady] = adz .
[adx, ady]w = adx adyw − ady adxw = [x, [y, w]] − [y, [x, w]] = [x, [y, w]] + [y, [w, x]] =
− [w, [x, y]] = [[x, y], w] = [z, w] = adzw .
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