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

1. Poisson algebraic Groups 13
Example 1.3.8. An important special case of the previous example is The
Kirillov-Kostant bracket. Let g a lie algebra. Consider the dual vector space
g ∗ equipped with the structure of abelian Lie group. The Kirillov-Kostant
bracket on g ∗ is given by:
{a, b} = [a, b] g
where a, b ∈ g are regarded as linear function on g ∗ .
Example 1.3.9. Consider the Lie group G = SL2(C) of complex 2 × 2
matrices with determinant 1. Then the following relations define a Poisson
Lie group structure on G:
{t11, t12} = −t11t12,
{t11, t21} = −t11t21,
{t12, t22} = −t12t22,
{t21, t22} = −t21t22,
{t12, t21} = 0,
{t11, t22} = −2t12t21,
where tij, (i, j = 1, 2), are the matrix elements.
1.3.3 The correspondence between Poisson Lie groups and Lie bialgebras
One of the most important facts of the Lie theory is the correspondence
between Lie groups and Lie algebras. Recall that given a Lie group G,
the tangent space at the unit element has a canonical Lie algebra structure.
Conversely, given a Lie algebra g, there exists a unique connected and simple
connected Lie group whose tangent space at the unit element is isomorphic
to g as Lie algebra.
We establish now a Poisson counterpart of this result. Let G a Poisson
Lie group, and g its Lie algebra. As usual identify g with the tangent space
TeG to G at the unit element of the group. Define a linear map φ : g → g ∧g
as the linear map which is dual to the Lie bracket in g ∗ = T ∗ e G given by
for any α, β ∈ g ∗ , where f, g ∈ C ∞ (G) are such that
[α, β] = de {f, g} (1.3)
def = α, deg = β.
Theorem 1.3.10. (i) Let G be a Poisson Lie group. Then there exists a
canonical Lie bialgebra structure on the Lie algebra g = TeG with the
cobracket φ : g → g ∧ g given by 1.3.
(ii) Let G be a Lie group, and suppose that the Lie algebra g = TeG is
equipped with a Lie bialgebra structure. Then there exist a unique Poisson
Lie group structure on G such that 1.3 holds.