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

4. General Theory 38
Proposition 4.1.5. (i) The elements Eβj , j = 1, . . .,k, generated a subalgebra
Uω which is independent of the choice of the reduced expression
of ω.
(ii) If ω ′ = ws with s a simple reflection and l(ω ′ ) = l(ω)+1 = k+1. then
Uω′ is a twisted polynomial algebra of type Uω
Eβk+1 σ,D , where σ and
D are given by the following formula, given in 4.6
Proof. See [DCP93].
= q (βj|βk+1) Eβj
σ Eβj
(4.8a)
D
Eβj = ckE k . (4.8b)
k∈Z N +
The elements Kα clearly normalize the algebra U ω and when we add
them to these algebras we are performing an iterated construction of Laurent
twisted polynomials. The related algebras will be called B ω .
4.1.2 Degenerations of quantum groups
We want to construct some degenerations of our algebra Uq(g) as the graded
algebra associated to suitable filtration.
Definition 4.1.6. Consider the monomials Mk,r,α = F kKαEr , where k =
(k1, . . .,kN), r = (r1, . . .,rN) ∈ ZN + and α ∈ Λ. The total height of Mk,r,α =
F kKαEr is defined by
d0(Mk,r,α = F k KαE r ) =
(ki + ri)htβi,
And its total degree by
d(Mk,r,α) = (kN, . . .,k1, r1, . . .,rN, d0) ∈ Z 2N+1
+ ,
where, ht β is the usual height of a root with respect to our choice of simple
roots.
We shall view Z 2N+1
+ as a total ordered semigroup with the lexicographic
order