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

5. Quantum universal enveloping algebras for parabolic Lie algebras 52
with h = |(R l ) + | and h + k = N = |R + |. We define, as in general case,
β 1 t = wsi1 . . .sit−1 (αit) ∈ (R l ) + ,
β 2
t = sj1 . . .sjt−1
αit+k
∈ R + \ (R l ) + .
Let U the quantum group associated to g define in 4.1.1, B the braid
group associated to W with the Lusztig action over U define by 4.5. Given
this choice, we obtain the q analogues of the root vectors:
and
E β 1 t
E β 2 t
F β 1 t
F β 2 t
= TwTi1
= Tj1
. . .Tit−1 (Eit),
. . .Tjt−1 .
Eit+k
= TwTi1 . . .Tit−1 (Fit) ,
= Tj1 . . .Tjt−1 .
Fit+k
The PBW theorem implies that, the monomials
E s1
β 2 1
· · ·E sk
β2E k
sk+1
β1 1
· · ·E sk+h
β1 KλF
h
tk+h
β1 h
· · ·F tk+1
β1 F
1
tk
β2 · · ·F
k
t1
β2 1
(5.1)
for (s1, · · · , sN), (t1, . . .,tN) ∈ (Z + ) N and λ ∈ Λ, form a C(q) basis of U.
The choice of the reduced expression of w0 and the LS relation for U
implies that
Property. For i < j one has
(i)
(ii)
E β 1 j E β 1 i − q (β1 i |β1 j ) E β 1 i E β 1 j =
k∈Z N +
ckE k 1
where ck ∈ C(q) and ck = 0 only when k = (s1, . . .,sk) is such that
sr = 0 for r ≤ i and r ≥ j, and Ek 1 .
= Es1
β 1 1
. . .E sk
β 1 k
E β 2 j E β 2 i − q −(β2 i |β2 j ) E β 2 i E β 2 j =
k∈Z N +
ckE k 2
where ck ∈ C(q) and ck = 0 only when k = (t1, . . .,th) is such that
tr = 0 for r ≤ i and r ≥ j, and Ek 2 .
The same statement holds for F β 1 i and F β 2 i .
= Et1
β 2 1
. . .E th
β 2 h
The definition of the braid group action implies: