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

5. Quantum universal enveloping algebras for parabolic Lie algebras 67
Lemma 5.4.22. Let θ = n i=1 aiαi the highest root of the root system R.
Let Z ′ = Z[a −1
1 , . . .,a−1 n ], and let Λ ′ = Λ ⊗Z Z ′ and Q ′ = Q ⊗Z Z ′ . Then the
Z ′ submodule (w0 − wl 0 )Λ′ of Q ′ is a direct summand.
Proof. The claim follows as a consequence of lemma 4.5.6.
So if l is a good integer, i.e. l is coprime with t and ai for all i, we have
rankT = l(w0) + l(w l 0) + n −
and so the theorem follows.
n − rank
5.5 The degree of Uǫ(p)
5.5.1 A family of Uǫ(p) algebras
w0 − w l 0
,
As we have seen at the end of section 5.4.1, Z0(p)[t] ⊂ Ut ǫ(p), so for all t ∈ C
and χ ∈ Spec(Z0(p)), we can define U t,χ
ǫ (p) = Ut ǫ(p)/J χ where Jχ is the two
side ideal generated by
kerχ = {z − χ(z) · 1 : z ∈ Z0(p)}
The P.B.W. theorem for U t ǫ(p) implies that
Proposition 5.5.1. The monomials
E s1
β 1 1
· · ·E sh
β1K h
r1
1 · · ·Krn n F tk+h
β1 h
· · ·F tk+1
β1 F
1
tk
β2 · · ·F
k
t1
β2 1
for which 0 ≤ sj, ti, rv < l, form a C basis for U t,χ
ǫ (p)
Lemma 5.5.2. For every 0 = λ ∈ C, U t,χ
ǫ (p) is isomorphic to U λt,χ
ǫ (p).
Proof. Consider the isomorphism ϑλ from U t ǫ(p) and U λt
ǫ (p), define by the
relation 5.10. Its follows from the above definition that ϑλ(J χ ) = J χ . Then
ϑλ induce an isomorphism between U t,χ
ǫ and U λt,χ
ǫ .
Proposition 5.5.3. The Uǫ(p) algebras U t,χ
ǫ (p) form a continuous family
parameterized by Z = C × Spec(Z0(p)).
Proof. Let V denote the set of triple (t, χ, u) with (t, χ) ∈ Z and u ∈ U t,χ
ǫ (p).
Then from the P.B.W. theorem we have that the set of monomial
E s1
β1
· · ·Esh
β1K h
r1
1 . . .Krn n F tN t1 · · ·F βN β1
for which 0 ≤ si, ti, rv < l, for i ∈ Π l , j = 1, . . .,N and v = . . .,n, form a
basis for each algebra U t,χ
ǫ .
Order the previous monomials and assign to u ∈ U t,χ
ǫ the coordinate
vector of u with respect to the ordered basis. This construction identifies V