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

(y).
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