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

subject to the following relations:
Proof. See [DCP93], §10.
4. General Theory 39
KαKβ = Kα+β, K0 = 1; (4.9a)
KαEβ = q (α|β) EβKα; (4.9b)
EαE−β = E−βEα if α, β ∈ R +
EαEβ = q
(4.9c)
(α|β) EβEα
E−αE−β = q (α|β) if α, β ∈ R
E−βE−α
+ and α > β. (4.9d)
Remark 4.1.8. a) Considering the degree by total height d0, we obtain a
Z+-filtration of U, let U (0) = Gr U the associated graded algebra. We
define by induction U (i) the graded algebra associated to U (i−1) with
respect to the Z+-filtration given by
rN−i+1 if 1 ≤ i ≤ N
di (Mk,r,α) =
ki−N if N + 1 ≤ i ≤ 2N
It is clear that at last step we get the algebra Gr U defined by 4.9, i.e.
U (2N) ∼ = Gr U
b) The algebra Gr U is a twisted polynomial algebra over C(q) on generators
Eβ1 , . . .,EβN , E−βN , . . . E−β1 , and K1, . . .,Kn, with the element Ki
inverted.
A first application of this methods is:
Theorem 4.1.9. The algebra U has no zero divisors.
Proof. Follows from remark 3.2.6.
4.2 Quantum groups at root of unity
To obtain from U a well defined Hopf algebra by specializing q to an arbitrary
non zero complex number ǫ, one can construct an integral form of U.
Definition 4.2.1. An integral form UA is a A subalgebra, where A =
C[q, q −1 ], such that the natural map
UA ⊗A C(q) ↦→ U
is an isomorphism of C(q) algebra. We define
Uǫ = UA ⊗A C
using the homomorphism A ↦→ C taking q to ǫ.