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

5. Quantum universal enveloping algebras for parabolic Lie algebras 55
Lemma 5.1.7. The map µ : U(p) → U(l) defined by
µ E s1
β1 · · ·E
1
sh
β1KλF h
tk+h
β1 · · ·F
h
tk+1
β1 F
1
tk
β2 · · ·F
k
t1
β2
1
0 if ti = 0 for some i = 1, . . .,h,
=
if ti = 0 for all i = 1, . . .,h,
E s1
β 1 1
· · ·E sh
β1KλF h
tk+h
β1 · · ·F
h
tk+1
β2 1
is an homomorphism of algebras.
Proof. Simple verification of the definition.
Proposition 5.1.8. U(p) and U(l) are Hopf subalgebra of U.
Proof. Follows immediately from the definition.
5.1.2 Definition of Uǫ(p)
Let A = C[q, q −1 ], and UA the integral form of U defined by 4.2.2. Like in
the general case, we define UA(p), has the subalgebra generated by E β 1 i , F β 1 i ,
F β 2 s , K ±1
j and Lj, with i = 1, . . .,h, s = 1, . . .,k and j = 1, . . .,n.
Definition 5.1.9. Let ǫ ∈ C, we define
Uǫ(p) = UA(p) ⊗A C
using the homomorphism A → C tacking q → ǫ
Let ǫ ∈ C such that ǫ 2di = 1 for all i, then
Property. Uǫ(p) ⊂ Uǫ(g). Moreover Uǫ(p) is generated by E β 1 i , Fβs and
K ±1
j , for i = 1, . . .,h, s = 1, . . .,N and j = 1, . . .,n.
Proof. The claim is a consequence of the definition of UA(p).
Proposition 5.1.10. The P.B.W. theorem and the L.S. relations holds for
Uǫ(p)
Proof. The claim is a consequence of the P.B.W. theorem and L.S. relation
for Uǫ(g) and the choice of the decomposition of the reduced expression of
w0.
5.2 The center of Uǫ(p)
Let ǫ ∈ C a primitive l th root of unity, with l odd and l > di for all i. Has
in section 4.2.1, we note that at root of unity the algebra Uǫ(p) has a big
center. The aim of this section is to extend some properties of the center of
Uǫ at the center of Uǫ(p).