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

1. Poisson algebraic Groups 9
Example 1.2.15. Let g be as in the previous example. Consider the Borel
subalgebras b±, note that they are in fact Lie bialgebras with respect to the
restriction of the cobracket from g to b±. Then the corresponding Manin
triple is
(g ⊕ h, b+, b−),
where the Borel subalgebras are embedded into g ⊕ h so that if a ∈ b+ then
a → (a, ah) and if a ∈ b− then a → (a, −ah), where we use the notation in
the previous example, the bilinear form is given by
〈(a, b), (c, d)〉 = 〈a, c〉g − 〈b, d〉h.
with a, c ∈ g and b, d ∈ h, and 〈, 〉h as the restriction of 〈, 〉g to h. In
particular, we see that the Borel subalgebras b+ and b− are dual to each
other as Lie bialgebra.
We consider, in the next example, an intermediate case between Borel
subalgebra b and Lie algebra g.
Example 1.2.16. Using the notation of the previous example, we call parabolic
subalgebra any subalgebra p of g such that b ⊆ p, note that b± and g
are examples of parabolic subalgebra. Set R as the set of rot associated to g
and Xα α ∈ R the root vectors, we define Rp := {α ∈ R : Xα ∈ p}. We call
l := h ⊕
α∈Rp:−α∈Rp CXα the Levi factor and u =
α∈Rp:−α/∈Rp CXα the
unipotent part of p. Moreover, we have p = l ⊕ u, for more details one can
see [Bou98] or [Hum78].
Proposition 1.2.17. p is a Lie bialgebra with respect to the restriction of
the cobracket from g to p.
Proof. Simple verification of the definition.
Proposition 1.2.18. The Manin triple corresponding to p is
(g ⊕ l, p, s),
where p is embedded into g ⊕ l so that a → (a, al) and
s = (x, y) ∈ b− ⊕ b l
+ : xh + yh = 0 ,
where b l + = b+ ∩ l, the bilinear form is given by
〈(a, b), (c, d)〉 = 〈a, c〉g − 〈b, d〉l.
with a, c ∈ g and b, d ∈ l, and 〈, 〉l is the restriction of 〈, 〉g to l.