1.3. POISSON-LIE GROUPS 9

The Poisson bivector field π is a section of T

2M

defined by

{f1,f2} = (df1 ∧ df2)(π).

In terms of the Poisson bivector field, a Poisson submanifold can be defined as a

submanifold N ⊂ M such that π|N ∈ T 2N. The rank of the Poisson structure at

a point of M is defined to be the rank of π at that point.

Now, we are ready to review the notion of a symplectic leaf. First, introduce an

equivalence relation ∼ on M: for x, y ∈ M, x ∼ y if x and y can be connected by a

piecewise smooth curve whose every segment is an integral curve of a Hamiltonian

vector field. Then it can be shown that

(i) equivalence classes of ∼ are Poisson submanifolds of M;

(ii) the rank of the Poisson structure at every point of such submanifold N is

equal to the dimension of N.

An equivalence class, Mx, of a point x ∈ M is called a symplectic leaf through

x. It is clear that Mx is a symplectic manifold with respect to the restriction of

{·, ·} to Mx. Casimir functions of {·, ·} are constant on symplectic leaves.

Example 1.3. An important example of a degenerate Poisson bracket is the

Lie-Poisson bracket on a dual space g∗ to a Lie algebra g defined by

{f1,f2}(a) = a, [df1(a),df2(a)]

for any f1,f2 ∈

C∞(g∗)

and a ∈

g∗.

Casimir functions for the Lie-Poisson bracket

are functions invariant under the co-adjoint action on

g∗

of the Lie group G = exp g.

Symplectic leaves of the Lie-Poisson bracket are co-adjoint orbits of G.

1.3.2. Let G be a Lie group equipped with a Poisson bracket {·, ·}. G is called

a Poisson-Lie group if the multiplication map

m : G × G (x, y) → xy ∈ G

is Poisson. This condition can be re-written as

(1.15) {f1,f2}(xy) = {ρyf1,ρyf2}(x) + {λxf1,λxf2}(y),

where ρy and λx are, respectively, right and left translation operators on G:

(ρyf)(x) = (λxf)(y) = f(xy).

Example 1.4. A dual space

g∗

to a Lie algebra g equipped with a Lie-Poisson

bracket (Example 1.3) is an additive Poisson-Lie group.

To construct examples of non-abelian Poisson-Lie groups, one needs a closer

look at the additional structure induced by property (1.15) on the tangent Lie

algebra g of G. First, we introduce right- and left-invariant differentials, D and D ,

on G:

(1.16) Df(x),ξ =

d

dt

f(exp (tξ)x)

t=0

, D f(x),ξ =

d

dt

f(x exp (tξ))

t=0

,

where f ∈

C∞(G),

ξ ∈ g. The Poisson bracket {·, ·} on G can then be written as

{f1,f2}(x) = π(x)Df1(x),Df2(x) = π (x)D f1(x),D f2(x) ,

where π, π map G into

Hom(g∗,

g). Then property (1.15) translates into the 1-

cocycle condition for π and π :

(1.17) π(xy) = Adx π(y) Adx−1

∗

+π(x), π (xy) = Ady−1 π (x) Ady

∗

+π (y).