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

5. Quantum universal enveloping algebras for parabolic Lie algebras 58
where Rap(Uǫ(p)) is the set of isomorphism classes of irreducible Uǫ(p) modules,
and Spec (Zǫ(p)) is the set of algebraic homomorphisms Zǫ(p) ↦→ C.
To see that Ξ is surjective, let I χ , for χ ∈ Spec (Zǫ(p)), be the ideal in
Uǫ(p) generated by
ker χ = {z − χ(z) · 1 : z ∈ Zǫ(p)} .
To construct V ∈ Ξ −1 (χ) is the same as to construct an irreducible representation
of the algebra U χ ǫ (p) = Uǫ(p)/I χ . Note that U χ ǫ (p) is finite dimensional
and non zero. Thus, we may take V , for example, to be any irreducible subrepresentation
of the regular representation of U χ ǫ (p).
Let χ ∈ Spec(Z0(p)), we define,
U χ ǫ (p) = Uǫ(p)/J χ
where J χ is the two sided ideal generated by
kerχ = {z − χ(z) · 1 : z ∈ Z0(p)}
5.4 A deformation to a quasi polynomial algebra
In this section we construct the main tool of this thesis. We want to modify
the relation that define the non restricted integral form of Uǫ(g), so as to
obtain a deformation of Uǫ(p) to a quasi-polynomial algebra. We begin by
constructing the deformation for Uǫ(g), which is exactly a reformulation of
the construction of Gr U in section 4.1.2.
5.4.1 The case p = g.
Definition 5.4.1. Let t ∈ C, we define Ut ǫ the algebra over C on generators
Ei, Fi, Li and K ± i , for i = 1, . . .,n, subject to the following relation:
K ±1
i K±1
j
KiK −1
= 1
i
= K±1
j K±1
i
Ki (Ej) K −1
i = ǫaij Ej
Ki (Fj)K −1
i = ǫ−aij Fj
⎧
[Ei, Fi] = tδijLi
⎪⎨ 1−aij adσ−α Ei Ej = 0
i
⎪⎩
1−aij adσ−α Fi Fj = 0
i
⎧
di −di ⎪⎨
ǫ − ǫ Li = t
⎪⎩
Ki − K −1
i
[Li, Ej] = t ǫaij −1
ǫdi−ǫ−di [Li, Fj] = t ǫ−aij −1
ǫdi−ǫ−di
EjKi + K −1
i Ej
FjKi + K −1
i Fj
(5.6)
(5.7)
(5.8)
(5.9)