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

5. Quantum universal enveloping algebras for parabolic Lie algebras 63
Set Sǫ(p) = U t=0
ǫ (p) ⊂ Sǫ.
Proposition 5.4.13. (i) For every t ∈ C, U t ǫ(p) is an Hopf subalgebra of
U t ǫ.
(ii) For any λ = 0, ϑλ defines by 5.10 is an isomorphism of algebra between
U t ǫ(p) and U λt
ǫ (p).
Proof. This is an immediate consequence of the same property in the case
p = g.
We can now state the main theorem of this section
Theorem 5.4.14. Sǫ(p) is a twisted derivation algebra
Proof. We use the same technique as we used on the proof of theorem 5.4.8.
Using the notation of section 4.5, let D(p) = Uǫ(b l +) ⊗ Uǫ(b−). Define
Σ : Sǫ(p) → D(p)
by Σ(Ei) = Ei, Σ(Fj) = Fj, Σ(K ±1
j ) = K ±1
j for i ∈ Π e l and j = 1, . . .,n.
Lemma 5.4.15. Sǫ(p) is a subalgebra of Sǫ
Proof. Note that D(p) is a subalgebra of D, and, as in lemma 5.4.4, the
map Σ is well define and injective. So, we have the following commutative
diagram
Σ
Sǫ(p)
D(p)
i
j
Σ
D
Sǫ
Since Σ and j are injective map, we have that i is also injective
So we can identify Sǫ(p) with the subalgebra of Sǫ generated by E β 1 i , Fβs
and K ±1
j for i = 1, . . .,h, s = 1, . . .,N and j = 1, . . .,n. As corollary of
proposition 5.4.6 and proposition 5.4.7, we have:
Proposition 5.4.16. (i) The monomials
E k1
β 1 1
. . . E kh
β1 K
h
s1
1 . . . Ksn n F t1 . . . Ft1
βN β1
for (k1, . . .,kh) ∈ (Z + ) h , (t1, . . .,tN) ∈ (Z + ) N and (s1, . . .,sn) ∈ Z n ,
form a C basis of Sǫ.
(ii) For i < j one has