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

5. Quantum universal enveloping algebras for parabolic Lie algebras 54
(ii) The monomials
E s1
β 1 1
· · ·E sh
β1KλF h
th
β1 · · ·F
h
t1
β1 1
for (s1, . . .,sh), (t1, . . .,th) ∈ (Z + ) h and λ ∈ Λ, form a C(q) basis of
U(l).
Proof. Follows easy from the P.B.W theorem for Uq(g).
Proposition 5.1.4. Set m = rankl = #|Πl |. The algebra U(p) is generated
by Ei, Fj Kλ, with i = 1, . . .,m, j = 1, . . .,n and λ ∈ Λ, subject to the
following relations:
KαKβ = Kα+β
(5.2)
K0 = 1
Where
n
m
KαEiK−α = q (α|αi) Ei
KαFjK−α = q −(α|αj) Fj
Kαi − K−αi
[Ei, Fj] = δij
qdi − q−di ⎧
⎪⎨
⎪⎩
di
1−aij
(−1)
s=0
s
1−aij
(−1)
s=0
s
1 − aij
s
1 − aij
s
di
di
E 1−aij−s
i EjEs i
= 0 if i = j
F 1−aij−s
i FjF s
i = 0 if i = j.
is the q binomial coefficient defined in section 3.1.
Proof. Follows from P.B.W. theorem and the L.S. relations.
We state now some easy properties of U(p):
Lemma 5.1.5. The multiplication map
is an isomorphism of vector spaces.
U + (l) ⊗ U 0 (l) ⊗ U − (l) → U(l)
Proof. Follows from the P.B.W theorem for U(l).
Lemma 5.1.6. The multiplication map
U(l) ⊗ U w m
−→ U(p)
(5.3)
(5.4)
(5.5)
defined by m(x, u) = xu for every x ∈ U(l) and u ∈ U w , is an isomorphism
of vector spaces.
Proof. The statement follows immediately from the proposition 5.1.3.