Student Seminar: Classical and Quantum Integrable Systems

where J is 2n × 2n matrix
and I is n × n unit matrix.
J =
( ) 0 I
−I 0
Linear representations of Lie groups Consider an action of a Lie group a n-
dimensional vector space R n . This action is called a linear representation of Lie
group G on R n if for any g ∈ G the map
ρ :
g → ρ(g)
is a linear operator on R n . In other words, by a linear representation of G on
R n we call the homomorphism ρ which maps G into GL(n, R), the group of linear
transformations of R n . The homomorphism means that under this map the group
structure is preserved, i.e.
ρ(g 1 g 2 ) = ρ(g 1 )ρ(g 2 ) .
Any Lie group G has a distinguished element – g 0 = I and the tangent space T at
this point. Transformation
G → G :
g → hgh −1
is called internal automorphism corresponding to an element h ∈ G. This transformation
leaves unity invariant: hIh −1 = I and it transforms the tangent space T into
itself:
Ad(h) : T → T .
This map has the following properties:
Ad(h −1 ) = (Adh) −1 , Ad(h 1 h 2 ) = Adh 1 Adh 2 .
In other words, the map h → Adh is a linear representation of G:
where n is the dimension of the group.
Ad : G → GL(n, R) ,
Generally, one-parameter subgroups of a Lie group G are defined as parameterized
curves F (t) ⊂ G such that F (0) = I and F (t 1 +t 2 ) = F (t 1 )F (t 2 ) and F (−t) = F (t) −1 .
As we have already discussed for matrix groups they have the form
F (t) = exp(At)
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