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

1. Poisson algebraic Groups 10
Proof. Observe that, the following decomposition hold in g
with u− :=
α∈R\Rp
, and
g = u− ⊕ l ⊕ u
dim(g ⊕ l) = dim(g) + dim(l) = dim(p) + u + dim(l) = 2 dim(p)
where u = dim u− = dimu+ Let us now check that p ∩ s = {0}, take
(a, b) ∈ p ∩ s, then
• (a, b) ∈ p ⇒ b = al
• (a, b) ∈ s ⇒ a ∈ b−, b ∈ b l + and ah + bh = 0
Then b = t ∈ h = b+ ∩ b−, and a = u + t with u ∈ u ⊂ b+, but a ∈ b−, so it
follows that u = 0 and a = b = t. Then the last condition gives us
0 = a + b = 2t ⇒ t = 0.
In conclusion we have a = b = 0. Hence p ∩ s = {0}. Observe that
then we have
dims = dimb+ + dimb l − − dimh = dimp,
g ⊕ l = p ⊕ s.
It remains to show that p and s are isotropic subspace with respect to 〈, 〉.
Recall that 〈, 〉g is an invariant symmetric bilinear form for g, note that
for any parabolic subalgebra b+ ⊆ q ⊆ g, its Levi factor m we have:
〈x, y〉g = 〈xh, yh〉m
for every x, y ∈ q.
Let (a, b) ∈ p then b = al, and we have
〈(a, b), (a, b)〉 = 〈a, a〉g − 〈b, b〉l
= 〈a, a〉g − 〈al, al〉l
= 0.
Let (a, b) ∈ s then ah + bh = 0. Recall that the Levi factor for b± is h. We
get
〈(a, b), (a, b)〉 = 〈a, a〉g − 〈b, b〉l
= 〈ah, ah〉h − 〈bh, bh〉h
= 〈ah, ah〉h − 〈−ah, −ah〉h
= 〈ah, ah〉h − 〈ah, ah〉h
= 0.
This finished the proof that (g ⊕ l, p, s) is the Manin triple associated to
p.