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

3. Twisted polynomial algebras 24
Definition 3.1.1. For integer 0 ≤ k ≤ n, set [0]! = 1,
if k > 0, and n
k
[k]! = [1] · · ·[k] ,
=
[n]!
[k]! [n − k]! .
Proposition 3.1.2. If x and y are variables subject to the following relation
xy = q 2 yx then, for n > 0,
(x + y) n =
n
k=0
q −k(n−k)
n
k
x k y n−k . (3.1)
Proof. We begin by stating the q analogue of the Pascal identity:
q k
n + 1
= q
k
n+1
n + 1 n
+
k − 1 k
then by induction on n the statement follow
Corollary 3.1.3. If q is a primitive l root of unity,and xy = q 2 yx then
Proof. Observe that
e
k
(x + y) e = x e + y e .
= 0 for all k such that 0 < k < e.
Apply this in the formula 3.1 and the statement fallow.
We give now some notations that will be useful in chapter 4 in order to
define the relations of the quantum groups. Fix d ∈ N, for all n ∈ Z, set
[n] d = qn − q−n qd .
− q−d We can now extend the definitions of q-factorial and q-binomial, in the following
way
Definition 3.1.4. For integer 0 ≤ k ≤ n, set [0]!d = 1,
if k > 0, and n
k
[k]!d = [1] d · · ·[k] d ,
d
[n]!d
= .
[k]!d [n − k]!d