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

1. Poisson algebraic Groups 8
Remark 1.2.12. Manin triple which corresponds to the double Lie bialgebra
is
(D(g) ⊕ D(g), D(g), g ⊕ g ∗ )
where D(g) is embedded into D(g) ⊕ D(g) as the diagonal, and g ⊕ g ∗ (the
dual Lie bialgebra of D(g)) is the direct sun of the Lie algebras g = (g, 0) ⊂
D(g) ⊕ D(g) and g ∗ = (0, g ∗ ) ⊂ D(g) ⊕ D(g). The bilinear form is given by
for any a, b, c, d ∈ D(g).
〈(a, b), (c, d)〉 = 〈a, c〉 D(g) − 〈b, d〉 D(g),
Double Lie bialgebra has an important property. Namely, the cobracket
on such Lie bialgebra is described by a very simple formula, as shown in the
following proposition
Proposition 1.2.13. Let g be a Lie bialgebra, and D(g) the double Lie
bialgebra. Suppose that {eα} is a basis of g and {e α } is the dual basis of g ∗ .
Let us identify eα with (eα, 0) ∈ D(g) and e α with (0, e α ) ∈ D(g). Then the
cobracket in D(g) is given by
for any a ∈ D(g), where
φ(a) = [a ⊗ 1 + 1 ⊗ a, r]
r =
is the canonical element related to 〈, 〉.
α
eα ⊗ e α ∈ D(g) ⊗ D(g),
We give now some important examples of Lie bialgebra and corresponding
Manin triple.
Example 1.2.14. Let g be a complex simple Lie algebra equipped with
the standard Lie bialgebra structure (cf. definition 1.2.5) related to a fixed
invariant bilinear form 〈, 〉, let h be the corresponding Cartan subalgebra
and define b± = n± ⊕ h a Borel subalgebra of g. Then the corresponding
Manin triple is (g ⊕ g, g, s), where g is embedded diagonally into g ⊕ g, and
s = {(x, y) ∈ b+ ⊕ b− : xh + yh = 0} ,
where xh denote the “Cartan part” of x ∈ b±. In other words, x = x± + xh
where x± ∈ n± and xh ∈ h. The invariant bilinear form g ⊕ g is given by
〈(a, b), (c, d)〉 = 〈a, c〉g − 〈b, d〉g.
Note that as Lie algebra D(g) is isomorphic to g ⊕ g. For more details one
can see for example [KS98].