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

Proof. Easy verification of the definition.
1. Poisson algebraic Groups 4
Let (M, π) be a Poisson manifold. Consider the morphism of vector
bundle ˇπ : T ∗ M → TM induced by π, we have
Definition 1.1.8. A Poisson manifold is called symplectic manifold if the
map ˇπ is an isomorphism.
1.1.2 Symplectic leaves
One of the most fundamental facts in the theory of Poisson manifolds is
that for any Poisson manifold M there is a stratification of M by symplectic
submanifolds which are called symplectic leaves in M. In a certain sense,
symplectic manifolds are simple objects in the category of Poisson manifolds.
In what follows we assume that M is a smooth Poisson manifold.
Definition 1.1.9. A Hamiltonian curve on a smooth Poisson manifold M
is a smooth curve γ : [0, 1] → M such that there exist f ∈ C ∞ (M) with the
property that
˙γ(t) = ξf(γ(t))
for any t ∈ (0, 1)
Definition 1.1.10. Let M be a smooth Poisson manifold.
1. We say that two points x, y ∈ M are equivalent if they can be connected
by a piecewise Hamiltonian curve.
2. An equivalence class of points of M is called symplectic leaf of M
Property. Let S be a symplectic leaf of a smooth Poisson manifold M.
Then:
(i) S is a Poisson submanifold of M;
(ii) S is a symplectic manifold;
(iii) M is the union of its symplectic leaves.
We need a tool to determine symplectic leaves.
Definition 1.1.11. Let P1 and P2 be Poisson manifolds, S a symplectic
f1
manifold. A diagram P1 ← S f2
→ P2 is called a dual pair, if f1 and f2
are Poisson maps and the Poisson subalgebras f ∗ 1 C∞ (P1) and f ∗ 2 C∞ (P2) of
C∞ (S) centralize each other with respect to the Poisson bracket.
A dual pair is called full if f1 and f2 are submersions.
f1
Theorem 1.1.12. Let P1 ← S f2
→ P2 be a full dual pair. Then the blow-up
−1
Mx1 = f2f1 (x1) is a symplectic leaf in P2 for any x1 ∈ P1.
Proof. See [KS98], page 8.