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

5. Quantum universal enveloping algebras for parabolic Lie algebras 53
Property. (i) If i ∈ Π l , then
for some s ∈ {1, . . .,h}.
(ii) If i ∈ Π \ Π l , then
for some s ∈ {1, . . .,k}.
Ei = E β 1 s ,
Fi = F β 1 s .
Ei = E β 2 s ,
Fi = F β 2 s .
Definition 5.1.1. The simple connected quantum group associated to p, or
parabolic quantum group, is the C(q) subalgebra of U(g) generated by
U(p) = 〈E β 1 i , Kλ, Fβj 〉
for i = 1, . . .,h, j = 1...N and λ ∈ Λ.
Definition 5.1.2. 1. The quantum Levi factor of U(p) is the subalgebra
generated by
U(l) = 〈E β 1 i , Kλ, F β 1 i 〉
for i = 1, . . .,h, and λ ∈ Λ.
2. The quantum unipotent part of U(p) is the subalgebra generated by
with s = 1...h
U w = 〈F β 2 s 〉
Set U + (p) = U + (l) = 〈Ei〉 i∈Π l, U − (p) = 〈Fi〉i∈Π, U − (l) = 〈Fi〉 i∈Π l and
U 0 (p) = U 0 (l) = 〈Kλ〉λ∈Λ. We have:
Property. The definition of U(p) and U(l) is independent of the choice of
the reduced expression of wl 0 and w0.
Proof. Follows immediately from proposition 4.1.5.
We can now state the P.B.W theorem for U(p) and U(l), which is an
immediately consequence of 5.1.
Proposition 5.1.3. (i) The monomials
E s1
β 1 1
· · ·E sh
β1KλF h
tk+h
β1 h
· · ·F tk+1
β1 F
1
tk
β2 · · ·F
k
t1
β2 1
for (s1, · · · , sh) ∈ (Z + ) h , (t1, . . .,tN) ∈ (Z + ) N and λ ∈ Λ, form a C(q)
basis of U(p).