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

5. Quantum universal enveloping algebras for parabolic Lie algebras 72
Then
E l i · u = E l iu − K l iuK −l
i El i = 0,
F l
i · u = F l
iuK −l l
i − uFiK −l
i = 0,
K l i · u = K l iuK −l
i = u.
It follows that U χ ǫ (p) is an U 0 ǫ (p) module.
Proposition 5.5.12. Let x ∈ U χ ǫ (p), then x is in the center of U χ ǫ (p) if and
only if x is invariant under the action of U 0 ǫ (p), i.e.
Proof. Let x ∈ Z (U χ ǫ (p)) then
Ei · x = 0, (5.25a)
Fi · x = 0, (5.25b)
Ki · x = x. (5.25c)
Ei · x = Eix − KixK −1
i Ei = 0,
in the same way we obtain the other relations.
Suppose now that x verify the relations 5.25. Then
Ki · x = KixK −1
i
= x,
imply that Kix = xKi. From Ei · x = 0 we obtain
0 = Ei · x = Eix − KixK −1
i Ei
= Eix − xEi.
its follows that Eix = xEi. In the same way we have Fix = xFi. Then x lies
in the center.
So we can determine the center at t generic by lifting the center of the
algebra at t = 0. We obtain an analogue of the Harish Chandra theory for
semisimple Lie algebras.