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

3. TWISTED POLYNOMIAL ALGEBRAS
In this chapter we introduce the main notion of quasi polynomial algebras,
or skew polynomial. Note that as the quantum enveloping algebras are the
“quantum” version of the universal enveloping algebras of a Lie algebra, we
can think that twisted polynomial algebras are the “quantum” version of
the symmetric algebra of a Lie algebra. More details on twisted polynomial
algebras can be found, for examples, in [DCP93] or [Man91].
3.1 Useful notation and first properties
Before giving the definition of twisted polynomial algebra, we want to introduce
some notations, all will be useful in the sequel.
Let fix an invertible element q ∈ C different from 1 and −1 so that the
fraction 1
q−q −1 is well defined. For all n ∈ Z, set
[n] = qn − q −n
q − q −1 = qn−1 + q n−3 + · · · + q −n+3 + q −n+1 .
We have the following relation:
[−n] = − [n]
[n + m] = q n [m] + q m [n]
Observe that if q is not a root of unity then ∀ n ∈ Z, non zero, [n] = 0. If q
is a primitive lth root of unity, with l > 2, define
l if l is odd
e =
.
if l is even.
Now is easy to check that
Property. If q is a primitive l root of unity then
(i) [n] = 0 ⇔ n ≡ 0 mod e
(ii) [n] l = [n].
l
2
We can now define the q analogue of the factorials and of the binomial
coefficients