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

5. Quantum universal enveloping algebras for parabolic Lie algebras 61
Proposition 5.4.7. For i < j one has
1.
2.
EβjEβi − ǫ(βi|βj)
EβiEβj =
k∈Z N +
ckE k
(5.14)
where ck ∈ C and ck = 0 only when k = (k1, . . .,kN) is such that
ks = 0 for s ≤ i and s ≥ j, and E k = E k1
β1
. . . EkN
βN .
FβjFβi − ǫ−(βi|βj)
FβiFβj =
k∈Z N +
ckF k
(5.15)
where ck ∈ C and ck = 0 only when k = (k1, . . .,kN) is such that
ks = 0 for s ≤ i and s ≥ j, and F k = F kN
βN
. . . Fk1
β1 .
Proof. We have by definition Eβi = Eβi ⊗ 1, then
EβjEβi − ǫ(βi|βj)
EβiEβj =
⎛
=
⎜
⎝ ckE k
⎞
⎟
⎠ ⊗ 1
Eβj Eβi − ǫ(βi|βj) Eβi Eβj
k∈Z N +
⊗ 1
where we have been using the L.S. relation for the Eβi . Note now that
then
E k = E k ⊗ 1,
EβjEβi − ǫ(βi|βj) EβiEβj =
⎛
⎜
⎝ ckE k
⎞
⎟
⎠ ⊗ 1
=
k∈Z N +
k∈Z N +
ckE k
In the same way, we can prove the L.S. relation for the Fβi
Theorem 5.4.8. (i) Sǫ = U t=0
ǫ is a twisted derivation algebra.
(ii) Gr Uǫ, where Gr Uǫ is defined by the relation 4.9, is a degeneration, in
the sense of twisted derivation algebra, of Sǫ
Proof. Define U0 = C[Eβ1 , FβN ] ⊂ Sǫ, then we can define
U i = U i−1
Eβi σ,D , FβN−i ⊂ Sǫ
where σ and D are given by the L.S. relation. Note now that, the Ki, for i =
1, . . .,n normalize U N , and when we add them to this algebra we perform an