8 1. PRELIMINARIES
The resulting factorization for X = AD is exactly formula (1.12) corresponding to
the word j = (−(n 1), −(n 2),..., −1, −(n 1), −(n 2),..., −2,..., −(n
1),n 1,n 2,n 1,..., 1,...,n 1, i, . . . , i(n 1)).
1.3. Poisson-Lie groups
The theory of Poisson-Lie groups and Poisson homogeneous spaces serves as
a source of many important examples that will be considered below. This section
contains a brief review of the theory.
1.3.1. We first recall basic definitions of Poisson geometry.
A Poisson algebra is a commutative associative algebra F equipped with a
Poisson bracket defined as a skew-symmetric bilinear map {·, ·} : F × F F that
satisfies, for any f1,f2,f3 F, the Leibniz identity
{f1f2,f3} = f1{f2,f3} + {f1,f3}f2
and the Jacobi identity
{f1, {f2,f3}} + {f2, {f3,f1}} + {f3, {f1,f2}} = 0.
An element c F such that {c, f} = 0 for any f F is called a Casimir element.
A smooth real manifold M is called a Poisson manifold if the algebra C∞(M)
of smooth functions on M is a Poisson algebra. In this case we say that M is
equipped with a Poisson structure.
Let (M1, {·, ·}1) and (M2, {·, ·}2) be two Poisson manifolds. Then M1 × M2 is
equipped with a natural Poisson structure: for any f1,f2 C∞(M1 × M2) and any
x M1, y M2
{f1,f2}(x, y) = {f1(·,y),f2(·,y)}1(x) + {f1(x, ·),f2(x, ·)}2(y).
Let F : M1 M2 be a smooth map; it is called a Poisson map if
{f1 F, f2 F }1 = {f1,f2}2 F
for every f1,f2 C∞(M2). A submanifold N of a Poisson manifold (M, {·, ·}) is
called a Poisson submanifold if it is equipped with a Poisson bracket {·, ·}N such
that the inclusion map i : N M is Poisson.
A bracket {·, ·} on C∞(M) is called non-degenerate if there are no non-constant
Casimir elements in C∞(M) for {·, ·}. Otherwise, {·, ·} is called degenerate. Every
symplectic manifold, that is, an even-dimensional manifold M equipped with a
non-degenerate closed 2-form
ω2,
gives rise to a nondegenerate Poisson bracket
(1.14) {f1,f2} =
ω2(Idf1,Idf2),
where I : T
∗M ω1

Iω1
TM is an isomorphism defined by
ω1,
· =
ω2(
· ,
Iω1).
Conversely, if a Poisson bracket {·, ·} on M is nondegenerate, then M is a symplectic
manifold and the Poisson bracket is defined by (1.14).
Fix a smooth function ϕ on a Poisson manifold M. By the Leibniz rule, the
map {ϕ, ·} :
C∞(M)

C∞(M)
is a differentiation. This implies an equation
{ϕ, f} = df, ,
which defines a vector field even if the Poisson bracket on M is degenerate.
is called a Hamiltonian vector field generated by ϕ. If M is a symplectic manifold,
then = Idϕ.
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