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

5. Quantum universal enveloping algebras for parabolic Lie algebras 65
Then the claim is proved.
Theorem 5.4.19. If l is a good integer (cf theorem 4.5.7)
deg Sǫ(p) = l 1
2(l(w0)+l(w l 0)+rank(w0−w l 0))
Proof. Denote by Sǫ(p) the quasi polynomial algebra associated to Sǫ(p).
We know by the general theory that
Let xi denote the class E β 1 i
deg Sǫ(p) = deg Sǫ(p).
for i = 1, . . .,h and yj the class of Fβj for
j = 1, . . .,N, then from theorem 5.4.14 we have
xixj = ǫ (β1 i |β1 j ) xjxi, (5.18)
yiyj = ǫ −(βi|βj) yjyi. (5.19)
if i < j. Thus we introduce the skew symmetric matrices A = (aij) with
aij = (βi|βj) for i < j and Al
= a ′
ij
with a ′ ij = (β1 i |β1 j ) for i < j.
Let ki the class of Ki, using the relation in theorem 5.4.14 we obtain a
n × N matrix B = ((wi|βj)) and a h × N matrix Bl = ((wi|β1 j )).
Let t = 2 unless the Cartan matrix is of type G2, in which case t = 6.
Since we will eventually reduce modulo l an odd integer coprime with t,
we start inverting t. Thus consider the free Z
1 module V + with basis
t
u1, . . .,uh, V − with basis u ′ 1 , . . .,u′ N and V 0 with basis w1, . . .,wn. On
V = V + ⊕ V 0 ⊕ V − consider the bilinear form given by
⎛
T = ⎝
A l − t B l 0
B l 0 −B
0 t B −A
then the rank of T is the degree of Sǫ(p). Consider the operators M l =
A l − t B l 0 , M = 0 t B −A , and N = B l 0 −B , so that
T = M l ⊕ N ⊕ M.
Note that
and
B(u ′ i) = βi
B l (ui) = β 1 i
Set T1 = Ml ⊕ M, then using the notation of lemma 4.5.4, we have
Lemma 5.4.20. The vector vω =
t∈Iω(wl 0 ) ut − ω − w0(ω) +
t∈Iω(w0) u′ t,
as ω runs through the fundamental weights, form a basis of the kernel of T1.
⎞
⎠,