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

1. Poisson algebraic Groups 12
1.3.2 Poisson Lie group
The definition of Poisson algebraic group formulated in the language of points
can be easily carried over to the case of Poisson Lie groups. The principal
difficulty is that the comultiplication maps goes into C ∞ (G × G) but not
necessarily into C ∞ (G) ⊗ C ∞ (G) = C ∞ (G × G)
Definition 1.3.3. Let G be a Lie group and at the same time, a Poisson
manifold. We say that G is a Poisson Lie Group if the multiplication m :
G × G → G is a Poisson map.
Note. Poisson Lie groups form a category, with morphisms being homomorphisms
which are the same time Poisson maps.
Proposition 1.3.4. Let G be both a Lie group and a Poisson manifold, and
let π be the bivector field corresponding to the Poisson manifold structure on
G. Then G is a Poisson Lie group if and only if π satisfies the following
condition:
π(g1g2) = (lg1 )∗π(g2) + (rg2 )∗π(g1)
for any g1, g2 ∈ G, where lg is the left multiplication and rg is the right
multiplication.
Proof. See [KS98], page 19.
Corollary 1.3.5. The unit element of a Poisson Lie group is always a zerodimensional
symplectic leaf
In this thesis we always used Poisson algebraic groups so we shall describe
Poisson brackets on the algebras of regular functions.
Example 1.3.6. Every Lie group G with trivial Poisson bracket is a Poisson
Lie group.
Example 1.3.7. Consider the abelian Lie group G = C n . Note that by
linearity it suffices to define the Poisson bracket on the coordinate function.
To get a Poisson Lie group structure we can take:
{xi, xj} =
n
k=1
c k i,jxk
where xm, m = 0...n, are the coordinate function, and the structure constant
ck i,j satisfy the following condition:
c k i,j = −c k j,i
n
c l i,jc m l,k + clj,k cml,i + clk,i cm
l,j = 0
l=1