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

Proof. See [KS98], page 23.
Proposition 1.4.2. The multiplication maps
1. Poisson algebraic Groups 15
m : G × G ∗ → D(G), (1.4)
m : G ∗ × G → D(G) (1.5)
are local Poisson diffeomorphisms in a neighborhood of the unit element.
Proof. This is an easy consequence of the fact that D(g) = g ⊕g ∗ as a vector
space.
It is clear that one can identify locally G with G ∗ \D(G) or with D(G)/G ∗ .
Consider the natural projection p1 : D(G) → G ∗ \ D(G) and p2 : D(G) →
D(G)/G ∗ . Then we have the following property
Proposition 1.4.3. The Poisson manifold structure on the double Lie group
D(G) induces Poisson manifold structure on G ∗ \ D(G) and D(G)/G ∗ such
that the natural projections p1 and p2 are Poisson maps. Both manifolds are
isomorphic to G as Poisson manifolds in a neighborhood of the coset G ∗ .
Now we introduce the notion of left and right dressing actions. First we
define them locally. Given g ∈ G and h ∈ G∗ which lie in some neighborhoods
of the unit element of D(G), using proposition 1.4.2 there exist unique gh ∈ G
and hg ∈ G∗ such that
hg = g h h g
(1.6)
This formula defines a local left action of G∗ on G and a local right action
of G on G∗ given by
h : g ↦→ g h , g : h ↦→ h g
(1.7)
respectively. Note that we can replace G by G ∗ and vice versa, so that we
have also a local right action of G ∗ on G and a local left action of G on G ∗ .
Definition 1.4.4. The local left (risp. right) action of G on G ∗ define by
1.6 and 1.7 is called local left (risp. right) dressing action of G on G ∗ . If
it can be extended to a global action, the latter is called global left (resp.
right) dressing action of G on G ∗ .
Proposition 1.4.5. Let G a Poisson Lie group and G ∗ the Poisson Lie
group dual to G. Then the symplectic leaves in G locally coincide with the
orbits of the right (or left) dressing action of G ∗ . In particular, if the right
(resp. left) dressing action is defined globally, the symplectic leaves are the
orbits of the right (risp. left) dressing action.
Proof. See [KS98], page 26.