1.3. POISSON-LIE GROUPS 11

R is called a classical R-matrix.

The invariant bilinear form (·, ·) can be represented by a Casimir element t ∈

g ⊗ g. Let r be the image of R under the identification g ⊗ g

∼

= g ⊗

g∗

∼

= End(g) and

let rij (1 ≤ i j ≤ 3) denote the image of r under the embedding g ⊗ g → g ⊗ g ⊗ g

such that the first (resp. second) factor in g ⊗ g is mapped into the ith (resp. jth)

factor in g ⊗ g ⊗ g. Define

[[r, r]] := [r12,r13] + [r12,r23] + [r13,r23].

Then condition (1.20) in the definition of a factorizable Lie bialgebra can be equiv-

alently re-phrased as a condition on the cobracket δ:

δ(ξ) = [ξ ⊗ 1 + 1 ⊗ ξ, r±],

where r± =

1

2

(r ± t) ∈ g ⊗ g satisfy

[[r±,r±]] = 0.

Let G be a Poisson-Lie group with a factorizable tangent Lie bialgebra (g,

g∗).

The corresponding Poisson bracket whose existence is guaranteed by Theorem 1.5

is called the Sklyanin bracket. To provide an explicit formula for the Sklyanin

bracket, we first define, in analogy to (1.16), right and left gradients for a function

f ∈

C∞(G):

(1.21) (∇f(x),ξ) =

d

dt

f(exp (tξ)x)

t=0

, (∇ f(x),ξ) =

d

dt

f(x exp (tξ))

t=0

.

Then the Sklyanin bracket has a form

(1.22) {f1,f2} =

1

2

(R(∇ f1), ∇ f2) −

1

2

(R(∇f1), ∇f2).

1.3.4. Standard Poisson-Lie structure on a simple Lie group. We now

turn to the main example of a Poisson-Lie group to be used in this exposition. Let

G be a connected simply-connected Lie group with the Lie algebra g. The Killing

form is a bilinear nondegenerate form on g

∼

=

g∗.

Define R ∈ End(g) by

(1.23) R(ξ) = ξ+ − ξ−,

where ξ± are defined as in (1.10). It is easy to check that R satisfies (1.20) and thus

(1.22), (1.23) endow G with a Poisson-Lie structure, called the standard Poisson-Lie

structure and denoted by {·, ·}G. Furthermore, the Lie bracket (1.19) on

g∗

∼

=

g in

this case is given by

(1.24) [ξ, η]∗ = [ξ+ +

1

2

ξ0,η+ +

1

2

η0] − [ξ− +

1

2

ξ0,η− +

1

2

η0].

It follows from (1.24), that subalgebras n± of g are ideals in

g∗.

Since n± =

b±⊥

with respect to the Killing form, Proposition 1.6 implies that Borel subgroups B±

are Poisson-Lie subgroups of G.

Example 1.7. Let us equip G = SLn and g = sln with the trace-form

(ξ, η) = tr ξη.

Then the right and left gradients (1.21) are

∇f(x) = x gradf(x), ∇ f(x) = gradf(x) x,