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

4. GENERAL THEORY
We briefly recall the notations introduced in chapter 1. Let g be a simple
Lie algebra, h a fixed Cartan subalgebra, and let b be a Borel subalgebra
of g such that h ⊂ b. We denote by C = (ai,j) the Cartan matrix of g,
so there exist di such that (diaij) is a positive symmetric matrix. Let R
be the associated finite reduced root system, Λ its weight lattice and Q its
roots lattice, W the Weyl group. The choice of b gives us a set of positive
root R + , a set of simple roots Π ⊂ R + and a set of fundamental weights
w1, . . .,wn ∈ Λ.
4.1 Quantum universal enveloping Algebras
The quantum groups which will be the object of our study arise as qanalogues
of the universal enveloping algebra of our semisimple Lie algebra
g.
Definition 4.1.1. A simply connected quantum group Uq(g) associated to
the Cartan matrix C is an algebra over C(q) on generators Ei, Fi (i =
1, · · · , n), Kα α ∈ Λ, subject to the following relations
KαKβ = Kα+β
(4.1)
K0 = 1
Where
n
m
σα (Ei) = q (α|αi) Ei
σα (Fi) = q −(α|αi) Fi
Kαi − K−αi
[Ei, Fj] = δij
qdi − q−di ⎧
⎪⎨
⎪⎩
di
1−aij
(−1)
s=0
s
1−aij
(−1)
s=0
s
1 − aij
s
1 − aij
s
di
di
E 1−aij−s
i EjEs i
= 0 if i = j
F 1−aij−s
i FjF s
i = 0 if i = j.
is the q binomial coefficient defined in section 3.1.
(4.2)
(4.3)
(4.4)
Note. When there is no possible confusion, we will simply denote Uq(g) by
U.