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]) = 1 + 1 ξ, δ(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];
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