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, Vϕ ,

which defines a vector field Vϕ even if the Poisson bracket on M is degenerate. Vϕ

is called a Hamiltonian vector field generated by ϕ. If M is a symplectic manifold,

then Vϕ = Idϕ.