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

3. Twisted polynomial algebras 27
3.3 Representation theory of twisted derivation algebras
We want to analyze some interesting cases of the previous constructions for
which the resulting algebras are finite modules over their centers and thus we
can develop for them the notion of degree and a good representation theory.
Let us first make a reduction, consider a finite dimensional semisimple
algebra A over an algebraic closed field k, let
i kei be the fixed points of
the center of A under σ where the ei are the central idempotents. We have
D(ei) = D(e 2 i ) = 2D(ei)ei,
hence D(ei) = 0. It follows that, decomposing A = ⊕iAei, each component
Aei is stable under σ and D and thus we have
Aσ,D[x] =
(Aei) σ,D [x].
i
This allows us to restrict our analysis to the case in which 1 is the only
central idempotent.
The second reduction is described by the following:
Lemma 3.3.1. Consider the algebra A = k ⊕n with σ the cyclic permutation
of the summands and let D be a twisted derivation of this algebra relative to
σ. Then D is an inner twisted derivation.
Proposition 3.3.2. Let σ be the cyclic permutation of the summand of the
algebra k ⊕n . Then
(i) k ⊕n
σ x, x−1 is an Azumaya algebra of degree k over its center
k [xn , x−n ].
(ii) k ⊕n
σ x, x−1 ⊗k[xn ,x−n ] k x, x−1 is the algebra of n × n matrices over
k x, x−1 .
Proof. [DCP93] proposition 6.1, page 56.
Assume now that A is semisimple and that σ induces a cyclic permutation
of the central idempotents
Lemma 3.3.3. (i) A = Md(k) ⊕n .
(ii) Let D be a twisted derivation of A relative to σ. Then the pair (σ, D)
is equivalent to the pair (σ ′ , 0) where
Proof. [DCP93] lemma 6.2, page 57.
Summarizing we have
σ ′ (a1, a2, . . .,an) = (an, a1, . . .,an−1)