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

1. Poisson algebraic Groups 16
Theorem 1.4.6. Let g ∈ G belong to a sufficiently small neighborhood of
the unit element in G. The symplectic leaf in G passing trough g locally (in
some neighborhood of g) is the image of the double coset G ∗ gG ∗ ⊂ D(G)
under the natural projection D(G) → D(G)/G ∗ ⋍ G
Proof. See [KS98], page 27.
1.4.2 Symplectic leaves in simple complex Poisson Lie groups
Let G be a finite dimensional simple complex Lie group, and g the Lie algebra
of G. Suppose that g is equipped with the standard bialgebra structure
described in the example 1.2.5. It induces a Poisson Lie group structure on
G which is also called standard. Our goal is to describe the symplectic leaves
in G.
Let (g⊕g, g, s) the Manin triple associated to g, described in the example
1.2.14. It is easy to see that D(G) is isomorphic to G × G as Lie group and
we can choose G ∗ = {(g1, g2) : g1 ∈ B+, g2 ∈ B− and (g1)H(g2)H = e} as
dual Poisson group. It is also clear that G · G ∗ is dense and open in D(G).
Moreover, the multiplication map m : G × G ∗ → G · G ∗ is a covering space.
We conclude that the image ˜ G of the quotient map p : G → D(G)/G ∗ is
dense and open in D(G)/G ∗ . By theorem 1.1.12 and theorem 1.4.6 we get
Lemma 1.4.7. (i) Any symplectic leaf in G is a connected component of
the fiber p −1 (S), where S is a symplectic leaf in ˜ G
(ii) Any symplectic leaf in ˜ G is of the form ˜ G ∩ G ∗ gG ∗ /G ∗ for some g ∈
G ⊂ D(G).
Now we want to describe the symplectic leaves. It appears that they are
related to the Bruhat decomposition.
Let b± = n± ⊕ h be the Borel subalgebra of g, and B± the corresponding
Borel subgroup of G. Recall that the Weyl group W is generated by the
simple reflection si : h ∗ → h ∗ given by:
si(λ) = λ −
2(λ, αi)
αi
(αi, αi
where α1, . . .,αr are the simple root, and r = dim h is the rank of g. The
following results are well know (cf. [Ste68])
Proposition 1.4.8. The following decomposition of G holds:
G =
B±ωB±
ω∈W
It is called the Bruhat decomposition of G
(1.8)