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

5. Quantum universal enveloping algebras for parabolic Lie algebras 60
in the same way, we have KiFjK −1
i = ǫ−aij Fi. Finally we have
and
1−aij
r=0
(−1) r
[Ei, Fj] = EiFj − FjEi
= (Ei ⊗ 1)(1 ⊗ Fj) − (1 ⊗ Fj)(Ei ⊗ 1)
1 − aij
r
= Ei ⊗ Fj − Ei ⊗ Fj
= 0
ǫi
E 1−aij−r
i EjE r i
=
= 0
1−aij
r=0
(−1) r
1 − aij
r
in the same way, we can verify the third relation of 5.13.
ei
E 1−aij−r
i EjE r i ⊗ 1
Note that Σ is injective, then we can identify Sǫ with the subalgebra of
D generated by Ei, Fi and Ki, for i = 1, . . .,n. We define now the analogues
of the root vectors for Sǫ:
Definition 5.4.5. For all i = 1, . . .,N, let
(i) Eβi := Eβi ⊗ 1 ∈ Sǫ
(ii) Fβi := 1 ⊗ Fβi ∈ Sǫ
As a consequence of this we get a P.B.W theorem for Sǫ.
Proposition 5.4.6. The monomials
E k1 . . . EkN
β1 βN Ks1 1 . . . Ksn n F h1 . . . Fk1
βN β1
for (k1, . . .,kN), (h1, . . .,hN) ∈ (Z + ) N and (s1, . . .,sn) ∈ Z n , form a C basis
of Sǫ. Moreover
Sǫ = S − ǫ ⊗ S 0 ǫ ⊗ S + ǫ
where S + ǫ (resp. S− ǫ and S0 ǫ ) is the subalgebra generated by Eβi
and Ki).
Proof. This follows from the injectivity of Σ and proposition 4.5.2
(resp. Fβi
Note. Its is clear that Eβi is also the image of the element Eβi ∈ Ut ǫ, where
the Eβi are non commutative polynomials in the Ei’s by Lusztig procedure
([Lus93]). The same thing is true for Fβi and Fβi .
We see now, that the L.S. relation holds for Sǫ.