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

5. QUANTUM UNIVERSAL ENVELOPING ALGEBRAS FOR
PARABOLIC LIE ALGEBRAS
Let ǫ be a primitive l th root of the unity, we have seen in the previous chapter
how the degree of Uǫ(g) can be calculated. As we have seen in section 4.5,
De Concini, Kaç and Procesi prove that Uǫ(b) is a quasi polynomial algebra
and they calculate the degree reducing itself to calculation of the rank of
a matrix as we saw in chapter 4, theorem 4.5.7, they obtain the following
formula
deg Uǫ(b) = l 1
2 (l(ω0)+rank(1−ω0))
In this chapter, we want to generalize the previous formula at p parabolic
subalgebra of g. If we wanted to follow the chosen direction for Uǫ(g), we
would have to determine the center Uǫ(p). This seems to be a much greater
and at the moment more open problem. Unfortunately Uǫ(p) is not a quasi
polynomial algebra, therefore we cannot directly apply the theory developed
in chapter 3. So the main idea is to see Uǫ(p) as a flat deformation of a
suitable quasi polynomial algebra, and using it to calculate the degree.
We recall some notations. Let g be a simple lie algebra, fix a Borel subgroup
b ⊂ g and a Cartan subalgebra h ⊂ b ⊂ g. Let p be a parabolic
subalgebra in standard position i.e. b ⊂ p. Let p = l ⊕ u the Levi decomposition
of p, with l the Levi factor and u the unipotent part.
Let R be the finite reduced root system associated to g, R + the positive
roots and Π ⊂ R + the simple roots. Define (R l ) + = R l ∩R + and Π l = Π∩R +
the positive respectively the simple roots of l with respect to our choice. Set
C = (diaij) and C ′ the Cartan matrix associated to g and l, W the Weyl
group associated to g and Wl ⊂ W the subgroup associated to l, Λ and Λ l
the weight lattice and Q and Q l the root lattice of g and l respectively.
5.1 Parabolic quantum universal enveloping algebras
5.1.1 Definition of U(p)
Let w l 0 be the longest element of Wl and w0 the longest element of W. We
can choose a reduced expression w0 = sj1 . . .sjk si1 . . .sih , such that wl 0 =
. . .sjk ,
si1 . . .sih is a reduced expression for wl 0 . We set w = w0(w l 0 )−1 = sj1