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

3. Twisted polynomial algebras 26
for a suitable pair (σ ′ , D ′ ). In order to see this, it is better to make the
formulas explicit separately when b = 0 and when a = 1. In the first case
yc = axc = a(σ(c)x + D(c)) = a(σ(c))a −1 y + aD(c),
and we see that the new automorphism σ ′ is the composition Ad(a)σ, and
D ′ = aD, where Ad(a)(x) = axa −1 .
In the case a = 1, we have
yc = (x + b)c = σ(c)x + D(c) + bc = σ(c)y + D(c) + bc − σ(c)b,
so that σ ′ = σ and D ′ = D + adσ b. Summarizing we have
Proposition 3.2.5. Changing σ, D to Ad(a)σ, aD (resp. to σ, D + adσ b)
does not change the twisted derivation algebra up to isomorphism.
Remark 3.2.6. It is clear that if A has no zero divisors, then the algebra
Aσ,D[x] and Aσ[x, x −1 ] also have no zero divisors.
Given a twisted polynomial algebra Aσ,D[x], we can construct a natural
filtration given by
deg(p(x)) = n
where p(x) = anx n + . . . + a0, an = 0. The associated graded algebra is
clearly Aσ[x].
Definition 3.2.7. We shall say that the algebra Aσ[x] is a simple degeneration
of A σ,D [x].
Example 3.2.8. Let A be an algebra over a field k of characteristic 0, let
x1, . . .,xn be a set of generators of A and let Z0 be a central subalgebra of
A.
For each i = 1, . . .,n, denote by A i the subalgebra of A generated by
x1, . . .,xi, and let Z i 0 = Z0 ∩ A i . We assume the following three conditions
hold for each i = 1, . . .,n:
1. xixj = b ij xjxi + P ij if i > j, where b ij ∈ k, P ij ∈ A i−1 ,
2. σi(xj) = b ij xj for j < i define an automorphism of A i−1 .
3. Di(xj) = P ij for j < i.
We obtain
A i = A i−1
σi,Di [xi]
as twisted polynomial algebra. For every i, we may consider the twisted
polynomial algebra A i defined by the relation xixj = bijxjxi for j < i. We
call this the associated quasi polynomial algebra.
Theorem 3.2.9. Under the above assumptions, the quasi polynomial algebra
A = A n is obtained from A by a sequence of simple degenerations.
Proof. See [DCP93] theorem 5.3, page 56.