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

Property. We have
4. General Theory 44
∆(ei) = ei ⊗ 1 + k −1
1
⊗ ei
∆(fi) = fi ⊗ ki + 1 ⊗ fi
∆(ki) = ki ⊗ ki
Proposition 4.2.13. Z0 is a Hopf subalgebra of Uǫ, as are Z 0 0
and Z ≤0
0 = Z− 0 Z0 0 .
, Z≥0
0 = Z0 0 Z+ 0
It follows that Spec(Z0) inherits a Lie group structure from the Hopf
structure of Uǫ. In fact
Property. The formula
z, z ′ = lim
q→ǫ
zz ′ − z ′ z
l (q l − q −l )
defines a Poisson bracket on Z0 which gives to Spec Z0 the structure of a
Poisson Lie group.
In order to study in more details the Poisson structure of Z0, we must
introduce some extra structure. For every q ∈ C, define derivations ei and
fi of Uq by
e i(u) =
f i (u) =
E l i
F l
i
u
!,
, (4.14)
[l]qi
u
[l]qi !,
. (4.15)
Note that, if we specialize to q = ǫ, we obtain at first sight an indeterminate
result, since the El i and F l
i are central and [l]ǫi = 0. However, we
have
Proposition 4.2.14. On specializing to q = ǫi = ǫ di the formulas 4.14
induces well defined derivation of Uǫ. In fact, we have the following explicit
formulas:
ei(Ej) = 1
−aij
−aij
2 r
r=1
ǫi
F r
i FjF l−r
i
− F l−r
i
ei(Fj) = 1
l δi,j(ǫi − ǫ −1
i )l−2 (Kiǫi − K −1
i ǫ −1
i )El−1
i
ei(K ±1
j ) = ∓ 1
2l aij(ǫi − ǫ −1
i )lF l
iK ±1
j
and f is obtained from e
i i by using
Tω0e −1
iTω0 = f i
r
FjFi ,
where i ↦→ i is the permutation of the nodes of the Dynkin diagram of g such
that ω0(αi) = −α i .