10 1. PRELIMINARIES

Using the identification

Hom(g∗,

g)

∼

= g ⊗ g, we conclude from (1.17) that the

map δ : g → g ⊗ g defined by

(1.18) δ(ξ) =

d

dt

π(exp(tξ))

t=0

satisfies

δ([ξ, η]) = [ξ ⊗ 1 + 1 ⊗ ξ, δ(η)] − [η ⊗ 1 + 1 ⊗ η, δ(ξ)].

In other words, δ is a 1-cocycle on g with values in the g-module g ⊗ g.

Another consequence of (1.17) is that the Poisson operators π, π vanish at the

identity element e of G and, thus, any Poisson-Lie bracket is degenerate at the

identity. The linearization of {·, ·} at the identity equips g∗

∼

=

Te ∗G with a Lie

algebra structure:

[a1,a2]∗ = de{ϕ1,ϕ2},

where ϕi, i = 1, 2 are any functions such that deϕi = ai. Comparison with (1.18)

gives

[a1,a2]∗,ξ = a1 ∧ a2,δ(ξ) .

To summarize, if G is a Poisson-Lie group then the pair (g,

g∗)

satisfies the

following conditions:

(i) g and

g∗

are Lie algebras,

(ii) the map δ dual to the commutator [·, ·]∗ :

g∗

⊗

g∗

→

g∗

is a 1-cocycle on g

with values in g ⊗ g.

A pair (g,

g∗)

satisfying the two conditions above is called a Lie bialgebra and

the corresponding map δ is called a cobracket.

Theorem 1.5. (Drinfeld) If (g,

g∗)

is a Lie bialgebra and G is a connected

simply-connected Lie group with the Lie algebra g, then there exists a unique Poisson

bracket on G that makes G into a Poisson-Lie group with the tangent Lie bialgebra

(g,

g∗).

If H is a Lie subgroup of G, it is natural to ask if it is also a Poisson-Lie subgroup,

i. e. if H is a Poisson submanifold of G such that property (1.15) remains valid for

the restriction of the Poisson-Lie bracket on G to H. The answer to this question

is also conveniently described in terms of the tangent Lie bialgebra of G. Let h ⊂ g

be the Lie algebra that corresponds to H.

Proposition 1.6. H is a Poisson-Lie subgroup of G if and only if the annihi-

lator of h in

g∗

is an ideal in the Lie algebra

g∗.

1.3.3. Next, we discuss an important class of Lie bialgebras, called factorizable

Lie bialgebras.

A Lie bialgebra (g, g∗) is called factorizable if the following two conditions hold:

(i) g is equipped with an invariant bilinear form (·, ·), so that g∗ can be identified

with g via

g∗

a → ξa ∈ g : a, · = (ξa, ·);

(ii) the Lie bracket on g∗

∼

=

g is given by

(1.19) [ξ, η]∗ =

1

2

([R(ξ),η] + [ξ, R(η)]) ,

where R ∈ End(g) is a skew-symmetric operator satisfying the modified classical

Yang-Baxter equation (MCYBE)

(1.20) [R(ξ),R(η)] − R ([R(ξ),η] + [ξ, R(η)]) = −[ξ, η];