In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it's a set with a Lie algebra and a Lie coalgebra structure which are compatible.
It is a bialgebra where the comultiplication is skew-symmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1-cocycle, so that the multiplication and comultiplication are compatible. The cocycle condition implies that, in practice, one studies only classes of bialgebras that are cohomologous to a Lie bialgebra on a coboundary.
They are also called Poisson-Hopf algebras, and are the Lie algebra of a Poisson-Lie group.
Lie bialgebras occur naturally in the study of the Yang-Baxter equations.
A vector space
g
is a Lie bialgebra if it is a Lie algebra, and there is the structure of Lie algebra also on the dual vector space
g
∗
which is compatible. More precisely the Lie algebra structure on
g
is given by a Lie bracket
[
,
]
:
g
⊗
g
→
g
and the Lie algebra structure on
g
∗
is given by a Lie bracket
δ
∗
:
g
∗
⊗
g
∗
→
g
∗
. Then the map dual to
δ
∗
is called the cocommutator,
δ
:
g
→
g
⊗
g
and the compatibility condition is the following cocyle relation:
δ
(
[
X
,
Y
]
)
=
(
ad
X
⊗
1
+
1
⊗
ad
X
)
δ
(
Y
)
−
(
ad
Y
⊗
1
+
1
⊗
ad
Y
)
δ
(
X
)
where
ad
X
Y
=
[
X
,
Y
]
is the adjoint. Note that this definition is symmetric and
g
∗
is also a Lie bialgebra, the dual Lie bialgebra.
Let
g
be any semisimple Lie algebra. To specify a Lie bialgebra structure we thus need to specify a compatible Lie algebra structure on the dual vector space. Choose a Cartan subalgebra
t
⊂
g
and a choice of positive roots. Let
b
±
⊂
g
be the corresponding opposite Borel subalgebras, so that
t
=
b
−
∩
b
+
and there is a natural projection
π
:
b
±
→
t
. Then define a Lie algebra
g
′
:=
{
(
X
−
,
X
+
)
∈
b
−
×
b
+
|
π
(
X
−
)
+
π
(
X
+
)
=
0
}
which is a subalgebra of the product
b
−
×
b
+
, and has the same dimension as
g
. Now identify
g
′
with dual of
g
via the pairing
⟨
(
X
−
,
X
+
)
,
Y
⟩
:=
K
(
X
+
−
X
−
,
Y
)
where
Y
∈
g
and
K
is the Killing form. This defines a Lie bialgebra structure on
g
, and is the "standard" example: it underlies the Drinfeld-Jimbo quantum group. Note that
g
′
is solvable, whereas
g
is semisimple.
The Lie algebra
g
of a Poisson-Lie group G has a natural structure of Lie bialgebra. In brief the Lie group structure gives the Lie bracket on
g
as usual, and the linearisation of the Poisson structure on G gives the Lie bracket on
g
∗
(recalling that a linear Poisson structure on a vector space is the same thing as a Lie bracket on the dual vector space). In more detail, let G be a Poisson-Lie group, with
f
1
,
f
2
∈
C
∞
(
G
)
being two smooth functions on the group manifold. Let
ξ
=
(
d
f
)
e
be the differential at the identity element. Clearly,
ξ
∈
g
∗
. The Poisson structure on the group then induces a bracket on
g
∗
, as
[
ξ
1
,
ξ
2
]
=
(
d
{
f
1
,
f
2
}
)
e
where
{
,
}
is the Poisson bracket. Given
η
be the Poisson bivector on the manifold, define
η
R
to be the right-translate of the bivector to the identity element in G. Then one has that
η
R
:
G
→
g
⊗
g
The cocommutator is then the tangent map:
δ
=
T
e
η
R
so that
[
ξ
1
,
ξ
2
]
=
δ
∗
(
ξ
1
⊗
ξ
2
)
is the dual of the cocommutator.
