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

4. General Theory 36
Note. One should think of Ei and Fi as q-analogues of the Chevalley generators
of g.
Theorem 4.1.2. U has a Hopf algebra structure with comultiplication ∆,
antipode S and counit η defined by:
⎧
⎨ ∆(Ei) = Ei ⊗ 1 + Kαi ⊗ Ei
• ∆(Fi) = Fi ⊗ K−αi + 1 ⊗ Fi
⎩
∆(Kα) = Kα ⊗ Kα
⎧
⎨ S(Ei) = −Kαi
•
⎩
Ei
S(Fi) = −FiKαi
S(Kα) = K−α
⎧
⎨ η(Ei) = 0
• η(Fi) = 0
⎩
η(Kα) = 1
Proof. See [Lus93].
Note. The quantum group in the sense of Drinfel’d-Jimbo is the subalgebra
UQ over C(q) generated by Ei, Fi, K ±1
i = K±αi (i = 1, · · · , n), we call it
also adjoint quantum group. More generally, for any lattice M between Λ
and Q, we can define UM to be the quantum group generated by the Ei, Fi
(i = 1, · · · , n) and the Kβ with β ∈ M.
We denote by U + , U − and U 0 the C(q)-subalgebra of UM generated by
the Ei, the Fi and Kβ respectively. The algebras U + and U − are not Hopf
subalgebras as one immediately sees from theorem 4.1.2. On the other hand,
the algebra U ≥0 := U + U 0 and U ≤0 := U 0 U − are Hopf subalgebras and we
shall think to them as quantum deformation of the enveloping algebras U(b)
and U(b − ), we denote them Uq(b) and Uq(b − ).
In fact we are interested in the study of a common generalization of Uq(g)
and Uq(b), namely Uq(p) for b ⊆ p ⊆ g a parabolic subalgebra.
4.1.1 P.B.W. basis
First, following Lusztig [Lus93], we define an action of the braid group BW
(associated to W). Denote by Ti the canonical generators of BW, we define
the action as an automorphism of U, by the formulas:
TiKλ = K si(λ)
TiEi = −FiKi
(4.5a)
(4.5b)
TiFi = −K −1
i Ei (4.5c)
TiEj =
−adE (−ai,j)
i
where if ∆(x) = xj ⊗ yj, then ad(x)(y) = xjyS(yj).
(Ej) (4.5d)