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

5. Quantum universal enveloping algebras for parabolic Lie algebras 70
Note. As we see in section 3.5, since Uǫ(p) is a maximal order, Zǫ(p) is
integrally closed, so following example 2.2.4, we can make the following construction:
denote by Qǫ := Q(Zǫ(p)) the field of fractions of Zǫ(p), we have
that Q(Uǫ(p)) = Uǫ(p) ⊗ Zǫ(p) Qǫ is a division algebra, finite dimensional over
its center Qǫ. Denote by F the maximal commutative subfield of Q(Uǫ(p),
we have, using standard tool of associative algebra (cf [Pie82]), that
(i) F is a finite extension of Qǫ of degree m,
(ii) Q(Uǫ(p)) has dimension m 2 over Qǫ,
(iii) Q(Uǫ(p)) ⊗Qǫ F ∼ = Mm(F).
Hence, we have that
dim Q(Z0(p)) Qǫ = deg τ
dimQǫ Q(Uǫ(p)) = m 2
dim Q(Z0(p)) Q(Uǫ(p)) = l h+N+n
where, the first equality is a definition, the second has been pointed out
above and the third follows from the P.B.W theorem. Then, we have
l h+N+n = m 2 deg τ
with m = l 1
2(l(w0)+l(w l 0 )+rank(w0−w l 0)) , so
Corollary 5.5.9.
deg τ = l n−rank(w0−w l 0) .
5.5.3 The center of U χ ǫ (p)
To conclude we want to explain a method, inspired by the work of Premet
and Skryabin ([PS99]), which in principle allows us to determine the center
of U χ ǫ (p) for all χ ∈ Spec(Z0).
Let χ0 ∈ Spec(Z0(p)) define by χ0(Ei) = 0, χ0(Fi) = 0 and χ0(K ±1
i ), we
set U 0 ǫ (p) = U χ0
ǫ (p).
Proposition 5.5.10. U 0 ǫ (p) is a Hopf algebra with the comultiplication,
counit and antipode induced by Uǫ(p).
Proof. This is immediately since J χ0 is an Hopf ideal.
Proposition 5.5.11. let χ ∈ Spec(Z0(p))
(i) U χ ǫ (p) is an Uǫ(p) module, with the action define by
where ∆(a) = a (1) ⊗ a (2).
a · u = a (1)uS(a (2)),