Degree of Parabolic Quantum Groups - Dipartimento di Matematica ...

Proof. See [KS98], page 21.
1. Poisson algebraic Groups 14
Proposition 1.3.11. (i) The correspondence G ↦→ g = TeG established a
covariant functor from the category of the Poisson Lie groups to the
category of Lie bialgebras.
(ii) The correspondence establishes an equivalence between the full subcategory
of connected and simply connected Poisson Lie groups and the
category of Lie bialgebras.
Example 1.3.12. Let G be a Lie group with trivial Poisson structure then
g is a Lie bialgebra with trivial cobracket
Example 1.3.13. Let G be the Lie group with Kostant-Kirillov structure,
define in example 1.3.8, then g is the Lie bialgebra given in the example 1.2.4
Definition 1.3.14. Let G be a Poisson Lie group, g the Lie bialgebra of G.
1. A connected Poisson Lie group G ∗ which corresponds to the Lie bialgebra
g ∗ is called a dual Poisson Lie group.
2. A connected Poisson Lie group D(G) which corresponds to the double
Lie bialgebra D(g) is called a double Poisson Lie group of G
1.4 Symplectic leaves in Poisson groups
1.4.1 Symplectic leaves in Poisson Lie groups and dressing action
Now we use theorem 1.1.12 to describe symplectic leaves in a Poisson Lie
group. The double Poisson Lie group construction allows us to describe
the symplectic leaves locally as the orbits of the so-called dressing action.
Throughout the section the word “locally” means “in a neighborhood of the
unit element of the group”.
Let G be a Poisson Lie group, and g the Lie bialgebra of G. Recall that
we have a simple description of the dual Lie bialgebra structure on D(g) by
φ(a) = [a ⊗ 1 + 1 ⊗ a, r]
for any a ∈ D(g), where r =
α eα ⊗ e α , {eα} being a basis of g and {e α }
the dual basis of g ∗ .
Proposition 1.4.1. The Poisson bracket on D(G) is given by
{f1, f2} =
(δαf1δ α f2 − δ α f1δαf2) −
α
α
δ ′ αf1(δ α ) ′ f2 − (δ α ) ′ f1δ ′ αf2
where δα (resp. δ ′ α) is the right (resp. the left) invariant vector field on D(G)
which takes the value eα at the unit element of D(G), while δ α (resp. (δ α )’)
is the right (resp. the left) invariant vector field on D(G) which takes the
value e α at the unit element of D(G).