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

5. Quantum universal enveloping algebras for parabolic Lie algebras 62
iterated construction of twisted Laurent polynomial. The resulting algebra
will be called T . We now claim
Sǫ = T
Note that, by construction T ⊂ Sǫ, so we only have to prove that Sǫ ⊂ T .
Now note that
E k1 · · · EkN
β1 βN Ks1 1 · · · Ksn n F hN · · · Fh1 ∈ T
βN β1
for every (k1, . . .,kN), (h1, . . .,hN) ∈ (Z + ) N and (s1, . . .,sn) ∈ Z n . Then
by proposition 5.4.6 we have Sǫ ⊂ T .
The second part is obtained by using standard technique of quasi polynomial
algebra. Denote by Sǫ the quasi polynomial algebra associated to Sǫ.
It easy to see that Sǫ ∼ = Gr Uǫ.
We finish this section with some remarks on the center of Ut ǫ. Recall that
Ut ǫ is isomorphic to Uǫ for every t ∈ C∗ , hence Zt ǫ is isomorphic to Z1 ǫ = Zǫ.
For t = 0, we define C0 the subalgebra of Sǫ generated by El β , Fl β for β ∈ R+
and K ±l
j for j = 1, . . .,n and let Cǫ be the center of Sǫ. Let Z0[t] the trivial
deformation of Z0
Lemma 5.4.9. (i) ρ : Z0[t] → U t ǫ defined in the obvious way is an injective
homomorphism of algebra.
(ii) U t ǫ is a free Z0[t] module with base the set of monomials
E k1 · · ·EkN β1 βN Ks1 1 · · ·Ksn n F hN h1 · · ·F βN β1
for which 0 ≤ ki, sj, hi < l, for i = 1, . . .,N and j = 1, . . .,n.
Proof. (i) follows by definitions of Z0[t]. (ii) follows from the P.B.W theorem.
Lemma 5.4.10. Z0[t]/(t) ∼ = C0 and Z0[t]/(t − 1) ∼ = Z0.
Proof. Follows from the definitions of Z0[t], C0 and Z0
Proposition 5.4.11. Uǫ and Sǫ are isomorphic has Z0 modules.
Proof. Follows from the previous lemma.
We can now study the general case.
5.4.2 general case
Definition 5.4.12. Let Ut ǫ(p) be the subalgebra of Ut ǫ generated by Eβ1, i
Fβj and K±1 s for i = 1, . . .,h, j = 1, . . .,N and s = 1, . . .,n.