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

3. Twisted polynomial algebras 25
3.2 Definition
Let A be an algebra over an algebraic closed field k, and σ an automorphism
of A.
Definition 3.2.1. A twisted derivation of A relative to σ is a linear map
D : A → A such that:
∀ a, b ∈ A.
D(ab) = D(a)b + σ(a)D(b)
Definition 3.2.2. A twisted derivation D is called inner, if it exists in an
element a ∈ A such that:
and we denote it adσa.
D(b) = ab − σ(b)a
Property. Let a ∈ A and σ be an automorphism such that σ(a) = q 2 a where
q is a scalar. Then
(adσa) m (b) =
m
(−1) j q j(m−1)
j=0
m
j
a m−j σ j (b)a j
Corollary 3.2.3. Under the hypothesis of Property 3.2 we have:
if q is a primitive l-th root of 1.
σ
(adσa) e (x) = a e x − σ e (x)a e
Fix an automorphism σ of A and a twisted derivation D of A relative to
Definition 3.2.4. We define the twisted derivation algebra A σ,D [x] in the
indeterminate x to be the k-module A⊗kk[x] thought as formal polynomials
with multiplications defined by the rule:
xa = σ(a)x + D(a).
When D = 0, we will call it twisted polynomial algebra and we denote it by
Aσ[x].
Let us notice that if a, b ∈ A and a is invertible we can perform the
change of variables
y := ax + b
and we see that
Aσ,D[x] = Aσ ′ ,D ′[x],