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

5. Quantum universal enveloping algebras for parabolic Lie algebras 68
with Z × C d , where d = l h+n+N , thereby giving A a structure of an affine
variety.
Consider the vector bundle π : V → Z, (t, χ, u) → (t, χ). Note that the
structure constant of the algebra U t,χ
ǫ (p), as well as the matrix entries of the
linear transformations which define the action of Uǫ(p) relative to the basis,
are polynomial in χ and t. This means that the maps
µ : V ×Z V → V, ((t, χ, u), (t, χ, v)) ↦→ (t, χ, uv)
ρ : Uǫ × V → V, (x,(t, χ, u)) ↦→ (t, χ, x · u)
where (t, χ) ∈ Z, u, v ∈ U t,χ
ǫ (p) and x ∈ Uǫ(p), µ defined on V a structure of
vector bundle of algebra and ρ a structure of vector bundle of Uǫ(p) modules.
The fiber of π above (t, χ) is the Uǫ(p) algebra U t,χ
ǫ (p).
Note. If we fix χ ∈ Spec(Z0), we have from theorem 5.4.17 that the family
of algebra U t,χ
ǫ (p) is a flat deformation of algebra over Spec C[t].
5.5.2 Generically semisimplicity
Summarizing, if ǫ is a primitive l th root of 1 with l odd and l > di for all i,
we have proven the following facts on Uǫ:
• U t ǫ and Uǫ(p) are domains because Uǫ(g) it is,
• U t ǫ and Uǫ(p) are finite modules over Z0[t] and Z0 respectively (cf lemma
5.4.9 and proposition 5.2.2).
Since the L.S. relations holds for Uǫ(p) and U t ǫ (cf proposition 5.1.10), we can
apply the theory developed in section 4.1.2, and we obtain that Gr Uǫ(p) and
Gr U t ǫ are twisted polynomial algebra, with some elements inverted. Hence
all conditions of theorem 3.5.3 are verified, so
Theorem 5.5.4. U t ǫ and Uǫ(p) are maximal orders.
Therefore, Uǫ(p) ∈ Cm, i.e. is an algebra with trace of degree m.
Theorem 5.5.5. The set
Ω = {a ∈ Spec(Zǫ(p)), such that the corresponding semisimple
representation is irreducible}
is a Zariski open set. This exactly the part of Spec(Zǫ(p)) over which U χ ǫ (p)
is an Azumaya algebra of degree m.
Proof. Apply theorem 2.2.7, with R = Uǫ(p) and T = Zǫ(p).
Recall that Zǫ(p) is a finitely generated module over Z0(p). Let τ :
Spec(Zǫ(p)) → Spec(Z0(p)) the finite surjective morphism induced by the
inclusion of Z0(p) in Zǫ(p). The properness of τ implies the following