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

5. Quantum universal enveloping algebras for parabolic Lie algebras 71
(ii) U χ ǫ (p) is an U 0 ǫ (p) module, with the action induced by Uǫ(p).
Proof. (i) We must verify the relation 4.2.2 on the generators. For any
u ∈ U χ ǫ , we have
Then
Ei · u = Eiu + KiuS(Ei) = Eiu − KiuK −1
i Ei, (5.21a)
Fi · u = FiuKi − uFiKi, (5.21b)
Ki · u = KiuK −1
i . (5.21c)
[Ei, Fi] · u = EiFi · u − FiEi · u
= [Ei, Fi] uS K −1
i + KiuS ([Ei, Fi])
= LiuS K −1
i + KiuS (Li)
= Li · u.
(5.21d)
Now we verify the ǫ-Serre relation in the case of aij = −1, we have:
E 2 i Ej · u = E 2 i Eju + K 2 i KjuS(E 2 i Ej)
+E 2 i KjuS(Ej) + (1 + ǫ −2
i )KiEiEjuS(Ei)
+(1 + ǫ −2
i )KiEiKjuS(EiEj) + K 2 i EjuS(E 2 i ),
EiEjEi · u = EiEjEiu + KiEjEiuS(Ei) + EiKjEiu(S(Ej)
+KiKjEiuS(EiEj) + EiEjKiuS(Ei)
+KiEjKiuS(E 2 i ) + EiKjKiuS(EjEi) + K 2 i KjuS(EiEjEi),
EjE 2 i · u = EjE 2 i u + KjE 2 i uS(Ej)
+(1 + ǫ −2
i )EjKiEiuS(Ei) + (1 + ǫ −2
i )KjKiEiuS(EjEi)
Note that 1 + ǫ 2 i
+EjK 2 i uS(E 2 i ) + KjK 2 i uS(EjE 2 i ).
− [2]diǫ−1 i = 0, where [n]di = qn−q−n ǫdi −ǫ−d , then i
2
Ei Ej − [2] di EiEjEi + EjE 2 i · u = 0
(5.22)
(5.23)
(5.24)
All other relation can be obtain with similar calculus. So U χ ǫ (p) is an
Uǫ(p) module.
, and K±l
j are in the center of Uǫ(p) for i ∈ Πl and
j = 1, . . .,n, and that
(ii) Recall that E l i , F l j
∆(E l i) = E l i ⊗ 1 + K l i ⊗ E l i
∆(F l
i) = F l
i ⊗ K −l l
i + 1 ⊗ Fi ) = K±l
i ⊗ K±l
i
∆(K ±l
i